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Issue n° 3, J

International Assoaa

International Gr

ureau Gravimétrique InternationalInternational Geoid Service

Joint Bulletin

Newton’sBulletinBulletin

January2006

ociation of Geodesy andandravity Field Service

ISSN 1810-8555

Newton's Bulletin N. 3 Summary SECTION I - "Reviewed Papers" BGI Papers: Calibration of a 3-accelerometer inertial gravimetry system for moving gravimetry (B. de Saint-Jean, J. Verdun, H. Duquenne, J.P Barriot, J. Cali) Application of a least square spectral filter in correcting abnormal meteorological drift of a LaCoste-Romberg Gravimeter (Okechukwu K. Nwofor, T. Chidiezie Chineke) IGeS Papers: A gravimetric quasi-geoid evaluation in the Northern region of Algeria using EGM96 and GPS/Levelling (M.A. Meslem) Improvement of the gravimetric model of quasigeoid in Slovakia (M. Mojzeš, J. Janák, J. Papčo) Comparison among spherical harmonic synthesis methods for functionals of the gravity field (E. Fantino, S. Casotto) On the potential of wavelets for filtering and thresholding airborne gravity data (M. El-Habiby, M.G. Sideris) A comparative study of the geoid-quasigeoid separation term C at two different locations with different topographic distributions (M. Sadiq, Z. Ahmad) Outlier detection in CHAMP kinematic orbit data to be used in gravity field determination (C. Gruber) Latest geoid determinations for the Republic of Croatia (T. Bašić, Z. Hećimović) Terrain effect on gravity field parameters using different terrain models (Z. Hećimović, T. Bašić) Identifying sea-level rates by means of wavelet analysis of altimetry and tide gauge data (E.V. Rangelova, R.S. Grebenitcharsky, M.G. Sideris)

Abstract. A moving inertial gravimetric system isbeing developed, consisting of three high precisionaccelerometers measuring accelerations along threenon parallel axes. The signal delivered by eachaccelerometer is an electric current, the intensity ofwhich is proportional to the accelerationexperienced by the test mass of the accelerometricsensor. This sensor is also very sensitive totemperature variations which are continuouslymonitored by an internal temperature sensor. Thecurrent given by each accelerometer is transformedinto a voltage sampled at 31.25 Hz, that is onesample every 32 ms, while the temperature issampled at a rate of one sample every 4.096 s.Our aim is to carry out the calibration of this

system in order to derive the relationship betweeneach digitalized value given by the accelerometersand the actual acceleration, taking into accounttemperature variations. Our calibration systempermits to tilt simultaneously the threeaccelerometers above a point where gravity hasbeen precisely determined. Thus, the accelerometerscan sense any acceleration value between 0 and thevalue of gravity at the measuring point(accelerometer axis is then vertical).We discuss the results of the calibration by

looking at the residuals between observed valuesand those coming from different theoreticalcalibration functions. We particularly focus on theperturbing phenomena such as temperature ormisalignment of the sensitive axis.

Keywords. Vector gravimetry, calibration,accelerometers, least squares.

1 Introduction

Since about a decade, gravimetry has significantlyevolved and several techniques can now be usedwith different precisions and resolutions (Bruton,2000). There is however a gap affecting mediumresolutions, ranging from 5 to 100 km, which cannot

be filled by ground-based gravimetry or spacegravimetry (Boedecker, 1994, Verdun et al., 2003).The miniaturization of both accelerometers andGPS receivers made possible the design of smallsize apparatus for moving gravimetry. Suchapparatus are used to cover hard access continentalregions like mountains, margins, deserts or riverslike Amazon, with a resolution within the range 5 to100 km.Three accelerometers, coupled with a 4-antennae

GPS system can be used as a cheap and handyinstrument to measure the three spatial componentsof the gravity vector (vector gravimetry).Such a gravimetry system is now being developed

in our laboratory consisting of three high precisionaccelerometers (type QA3000-20, Honeywell)supported by a triad. Each accelerometer delivers anelectric current, the intensity of which isproportional to the acceleration sensed onto itssensitive axis. The current is then transformed into avoltage which is digitalized at a frequency up to250 Hz by means of a 24 bits A/D converter. Theinternal accelerometer temperature is also measuredby a sensor, and digitalized with a sampling rate of0.244 Hz (1 sample every 4.096 s). Our approachconsists of finding a calibration procedure in orderto estimate the relationship between the actualacceleration value and its digitalized value. Theeffects of temperature have also to be carefullyinvestigated since the electronic system is likely tobe very sensitive to temperature variations.We develop and discuss in this paper the

calibration procedure including the materials andthe processing of calibration function. A firstcalibration carried out in February 2005 is analysed,and a first calibration function is derived. We alsopropose some recommendations in order to improvethe reliability of the calibration procedure.

2 Calibration principle

Deriving the calibration function requires theacquisition of a set of accelerometer data at pointswhere the acceleration has been already measured

Calibration of a 3-accelerometer inertial gravimetrysystem for moving gravimetry

B. de Saint-Jean, J. Verdun, H. Duquenneemail : [email protected] Géographique National/LAREG, 6/8 av. Blaise Pascal, Champs sur Marne, 77455 Marne la vallée cedex 2, France

J.P. BarriotCentre National d'Études Spatiales (CNES), 18 av. E. Belin, 31401 Toulouse cedex 9, France

J. CaliÉcole Supérieure des Géomètres Topographes (ESGT), 1, Bd. Pythagore,72000 Le Mans, France

by means of another method. Our calibration systempermits to tilt simultaneously the threeaccelerometers above a point where gravity hasbeen measured beforehand using an absolutegravimeter. By so doing, each accelerometer cansense any acceleration value between 0(accelerometer sensitive axis is horizontal) and thegravity value at the point (980 855.8 mGal).Indeed, if the accelerometer is tilted by an angle α

from the vertical (fig. 1), the accelerometermeasurement F consists of the projection of gravityvector

gabs onto its sensitive axis, that isF = g

abs ∗ cos (1)

where β is the projection of θ (the angle between thegeometric axis and the sensitive axis of theaccelerometer) on the plan of α formed between thegeometric axis of the mirror and the sensitive axis ofthe accelerometer (fig. 2).

The angle θ is designed by the manufacturer to beless than 1 mrad. As a first approximation, we choseto neglect this angle because so far measurementsfor calibration do not permit to define this angle.So, the sensitive axis is assumed to be aligned withthe geometric axis of each accelerometer. Let D z bethe zenithal distance of the accelerometer axis(fig. 2), then we obtainF = g

abs ∗ cos D z . (2)

Equation (2) allows to determine the accelerationF sensed by each accelerometer for any zenithaldistance D z . In the meantime, the A/D converterprovides the digitalized value N g of the accelerationF. Our aim is now to obtain, for each accelerometer,a set of N observations (N g i , F i ), i = 1 , . . . , N , in orderto estimate the calibration function

A m = f T N g "(3)

of each accelerometer, for a given temperature.

Fig. 2 Measurement of D z3 Practical determination ofacceleration F

The three accelerometers are mounted on aplatform which can rotate around a horizontal axis,and enclosed in a thermally isolated box wheretemperature is maintained constant. The platform'srotation is controlled by a screw which can beovertightened in order to maintain the platform in afixed direction. The platform is also equipped with amirror placed outside the isolated box by means ofan axis. The axes of respectively the mirror and thethree accelerometers have been previously yieldedparallel by autocollimation with an optical plummet.By so doing, the zenithal distance of the mirror axiscorresponds exactly to the geometrical axes of thethree accelerometers. The zenithal distance can thenbe precisely measured by means of autocollimationusing a theodolite.

4 Result of the calibration

The calibration was carried out in the laboratory

Fig. 1 Tilt of accelerometer sensitive axis

Tilt from the vertical : αF g a b sAccelerometer

Sensitive Axis

θαD z αsensitive axis of

the accelerometer

α : tilt of the mirrorθ: misalignement of the sensitive axisD z : zenithal distance

Mirror forautocollimation

geometric axis ofthe accelerometerand the mirror

of ESGT (Ecole Supérieure des Géomètres etTopographes) during two days.

4.1 Acceleration measurements andprecision

Accelerations were measured for about 20zenithal distances D z ranging from 109° to 140°,and for two different temperatures. Wesystematically acquired about 15 measurements foreach zenithal distance, so as to estimate the standarddeviation D z . Then, assuming that the errors on D zand g a b s are uncorrelated, and the error on g a b s isnegligible, the standard deviation A m of F

can becalculated as: A m = g abs

sin D z D z . (4)

The resulting standard deviations have proved torange between 0.8 mGal and 4.5 mGal with amedian at 2.1 mGal.

4.2 Digitalizing of accelerations andprecision

For a given tilt, acceleration was continuouslydigitalized with a sampling rate of 31.25 Hz, i.e. onesample every 32 ms. By so doing, we acquired onaverage 80 digitalized values N g of F

for eachmeasurement at zenithal distance D z . As a result,using 15 zenithal distance measurements, weobtained 15 * 80 = 1200 digitalized values N g ofthe same acceleration. The standard deviations ofthe digitalized acceleration mean values rangebetween 0.5 and 2.5 bits with a median at 1.0 bit.Given the fact that accelerometers range extendsover 4 g (± 2 g ), coded on 24 bits, the resultingresolution is given by:

4 × 10 × 105224 = 0.24 m G a l / b i t

These findings indicate that the errors ondigitalized acceleration values cause an error onacceleration ranging between 0.12 mGal and0.60 mGal.The synchronisation of both the theodolite and the

A/D converter was ensured by means of GPS timedelivered by a dedicated GPS receiver.

5 Estimation of the calibration function

Following the manufacturer recommendation, the

calibration function for a given temperature has tobe chosen as a polynomial function. Since thetemperature does not vary very much and the rangeof accelerations is not wide, we chose a first orderpolynomial function as a model:f T N g = b k N g (5)

where b and k are the bias and the scale factor of thecalibration function, respectively.Let b 0 and k 0 be approximate values of parameters

b and k, then the calibration function for eachobservation (D z i , N g i ) may be expressed as g cos " D z i & v D z i + & " b 0 & - b + & " k 0 & - k + " N g i & v N g i + = 0where v D z i and v N g i correspond to the residuals of

the observations D z i and N g i , respectively, and 8 b ,8 k are the correction to be applied to the parametersb and k . By keeping only first order terms in theprevious equation, we obtaing cos " D z i + 9 g sin " D z i + v D z i& b 0 & - b & k 0 N g i & N g i - k & k 0 v N g i = 0 (7)

By denotingA = ; ;1 N g i; ; , X = - b- k , W = ;g cos D z i & b 0 & k 0 N g i;V = ;9 g sin D z i v D z i & k 0 v N g i;

equation 7 can be rewritten in matrix form asW & V & A X = 0 (9)

Estimates for b and k parameters can be found bymeans of the least squares method which consists inminimizing the quadratic form@ " V , A , X + = V T P V V 9 2 A T " W & V & A X + (10)

where P V is the weight matrix of residual vector Vand Λ is the vector of Lagrange's multipliers. Thesolution can be easily determined using thefollowing relations (Leick, 1990)

(6)

(8)

X = 9 " A T P V A + 1 A T P V WA = 9 P V " W & A X +V = P V 1 A (11)

The variance-covariance matrix of the parametersis the given by C X = " A T P V A + 1 (12)

The weight matrix P V can be deduced from theCv covariance matrix of vector V = [ v D zT , v N gT ] T

by(Fotopoulos, 2005)P V = " E C v E T + 1 w i t h

C v = v D z1 " 0 + v D z N v N g

1" 0 + v N g1E = 9 g sin " D z 1 + 9 k 0 9 g sin " D z N + 9 k 0 D z i and N g i for i = 1 , . . , N can be estimated from the

measurements described in § 4.1 and 4.2. Theabove-mentioned calculation can be performed byiteration starting from an arbitrary solution (b 0 , k 0 )and testing the sum of squared normalized residualsV T P V V

.The results obtained for our two-days calibration

are tabulated in table 1 for two differenttemperatures (24.8°C for the first day and 31.0°Cfor the second).

Table 1 Results of least squares estimation ; the secondaccelerometer was disconnected during the 2nd day

b k σb σk Somgal mgal/kbit mgal mgal/Mbit mgal

Day 1 acc. 1 -29497.03 243.05 4.58 1.83 0.03222 values acc. 2 -34476.42 241.77 4.62 1.82 0.03T°=24.8°C acc. 3 -23802.24 242.20 4.55 1.83 0.07Day 2 acc. 1 -29410.07 242.85 2.31 0.95 0.03

413 values acc. 2 --- --- --- --- ---T°= 31.0°C acc. 3 -24061.45 242.18 2.29 0.95 0.24

The relative uncertainties on the bias are verysmall, which gives confidence in its determination.

The scale factor has proved to be nearly constant(variation less than 2 mGal/kbits) for allaccelerometers and all temperatures which confirmsof the reliability of the accelerometer for trackingacceleration variations. The results suggest that thebias might increase with the temperature, which hasto be validated by exploring a wider range oftemperatures.

6 Conclusions

The calibration procedure proved to be veryefficient to estimate the calibration functions of eachindividual accelerometer. But, the misalignment θand its direction cannot be defined with thiscalibration process. Further experiments have to beheld in order to determine this angle.The bias is likely to be significantly influenced by

the temperature variations ; thus some additionalexperiments have to be carried out in order toexplore a wider temperature range.Clearly more work needs to be done particularly

in testing higher order calibration functions.

References

Boedecker, G., Leismüller, F., Spohnholtz, T., Cuno, J., andNeumayer, K. H. (1994). Accelerometer / gps integrationfor strapdown airborne gravimetry : First test results. InSunkel, H. and Marson, I., editors, G r a v i t y a n d G e o i d –I A G S y m p o s i u m , volume 113, pages 177–186. Springer.

Bruton, M. A. (2000). I m p r o v i n g t h e A c c u r a c y a n dR e s o l u t i o n o f S I N S / D G P S A i r b o r n e G r a v i m e t r y . PhD

thesis, University of Calgary.

Fotopoulos, G. (2005). Calibration of geoid error models viaa combined adjustment of ellipsoidal, orthometric geoidheight data. J o u r n a l o f G e o d e s y , 79 :11-123.

Leick, A. (1990). G P S S a t e l l i t e S u r v e y i n g John Wiley andSons (Eds).

Verdun, J., Klingelé, E., Bayer, R., Cocard, M., Geiger, A.,and Kahle, H.-G. (2003). The Alpine Swiss-Frenchairborne gravity survey. G e o p h y s i c a l J o u r n a lI n t e r n a t i o n a l , 152 :8–19.

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Application of a least square spectral filter in correcting abnormal meteorological drift of a LaCoste-Romberg Gravimeter

Okechukwu K, Nwofor and T Chidiezie, Chineke

Department of physics, Imo State University, P.M.B 2000 Owerri, Nigeria

ABSTRACT

The performance of a least square spectral filter in removing abnormal meteorological drifting earlier

reported for a Lacoste-Romberg gravimeter is evaluated. A short (3-days) drift curve of the instrument

was compared with tidal data for the location that is derived from theoretical Earth parameters before

and after application of the filter. The filter was effective in remove noise registrations above 4 cycles

per day in the gravimeter record. A large amplitude disparity at frequencies < 2 cycles per day between

the two time series were found after application of the filter. This is attributed to temperature induced

creeps in the meter spring, which resulted in phase shifts in the gravimeter series. It is concluded that a

combination of “Optimum Operating Procedures” as earlier published and appropriate filtering and

phase adjustments may increase gravimeter precision for measurements requiring mGal accuracy. µGal

precision data would however require instruments with better temperature compensation. The

procedure presented here is suggested for routine assessment of gravimeter precisions prior to field

deployment, against the background of scarcity of new gravimeters for fieldwork in Nigeria.

Key words: Spectral filter, least square, LCR gravimeter, meteorological drift, and synthetic Earth

tide.

1.0 INTRODUCTION

The Lacoste Romberg (LCR) gravimeter is useful in mapping geologic structures of anomalous

densities associated with ore and hydrocarbon accumulation (Ojo, 1992). In geodesy, it is useful in the

monitoring of elevation changes (Woollard, 1980) and surface deformations (Kiviniemi, 1974); and in

geodynamics for monitoring tectonic stresses (Honkassalo, 1975). One of the major advantages in

using the Lacoste-Romberg gravimeter for fieldwork is that it records minimal drift when stationary.

This facility is reduced as a consequence of aging and mechanical faults, (Osazuwa & Ajakaiye, 1982;

Nwofor, 1994; Nwofor & Chineke, 2003). Since the few available LCR gravimeters in Nigeria are

mostly very old and as such have lost most of the in-built compensations making them highly

susceptible to meteorological effects. There is the need to continue to assess the reliability of the

available instruments for various applications. Since synthetic tides of the Earth can be determined to a

very high level of precision (such as nano-Gals), via software: (http://www.gik.uni-

karlsruhe.de/Forschung/eterna33.htm), one of the problems presently encountered is to measure to this

accuracy using instruments. Synthetic Earth tide data can therefore provide a facility for routine

investigation of abnormal drifting of a gravimeter. This work introduces this method. It involves the

comparison of a short series of ground based gravity tidal observation of a LaCoste-Romberg

gravimeter in Jos Nigeria with a synthetic tide derived from wave groups that are based on theoretical

Earth parameters. The LCR gravimeter is among the very old ones commonly found in the country

whose several compensations were suspect, making it highly susceptible to meteorological influences.

“Optimum Operating Principles” for reducing these impacts some of which can interfere in the tidal

band, have been studied for the system as reported by Nwofor and Chineke (2003). The present method

is an appraisal of the effectiveness a low pass least square filter in reducing these abnormal drifts.

2. 0 METHODS

The gravimeter tide is a 72 hours ground based data beginning at 08,30 hours local time (UTC) on

October 10, 1992, obtained with a LaCoste Romberg (LCR) using the optical method. The gravimeter

model G.468 was positioned in Jos, Nigeria at a longitude of 8o 53’ E, latitude of 9o 57’ N, elevation of

1159 meters above sea level (Macleod et al., 1971); and about 1000 km from the southern coastline.

The gravimeter measures to 1 part in 100 million or 0.01 mGal (1 mGal =10-5 ms-2), which suites the

limit of sensitivity for the major tides, if instrumental and interference errors are removed. Two major

errors are however encountered with the use of the instrument. The first is due to spring response to

dial turns which may not be simply accurate and the other due to spring hysteresis. This second error

type is easily associated to loading, aging and meteorological effects.

The gravimeter tidal record is superimposed on the linear drift, which agrees in value with the normal

specification (0.0079 mGals/hour) (figure 1). The amplitudes h of the gravimeter series were then

obtained by fitting a linear trend d to the gravimeter readings G according to the equation

h = G – d (1)

Fig. 1a Fig 1a : Observed gravimeter drift (solid lines) w ith fitted linear trend (dashed

lines)

0

5000

10000

15000

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73

time/ hours

gra

vim

ete

r re

ad

ing

/

nm

s-2

Fig 1b

Figure 1: (a) Observed gravimeter drift (solid lines) with fitted linear trend (dashed lines)

(b) Gravimeter drift corrected for linear trend trend

The gravimeter data has a high signal to noise ratio, and therefore required appropriate filtering. Some

criteria for choosing frequencies to be removed have been explained by Mishra and Rao (1997). In

their work, temporal variations in the gravity field recorded at a station were classified into six major

types; very large period (100 – 10000 years); large period (10 – 100 years); medium period (days –

years); short period (hours – days); shorter period (hours) and shortest period (seconds – minutes). In

the present case the noise registrations are obviously shorter period events, which are found to be

mainly meteorological i.e. changes in atmospheric pressure (Durcame et al., 1999) and perhaps

fluctuations in the tropical temperatures (Nwofor 1994, Nwofor and Chineke, 2003).

We applied a low pass least square filter at a cut off frequency of 4 cycles per day in other to preserve

signals in the tidal band. The filter is implemented in the model tide programme Tsoft (Van Camp &

Vauterin, (2005)) which was also used to compute the synthetic tide for the station using a wave group

table.

The Least Square Spectral Filter (LSSF) is found to be particularly suitable for the data series. This is

because the Discrete Fourier Transform (DFT), which is an alternative (Scales, 1997), may not be

suitable for analyzing data with gaps in the series (Vanicek, 1971), and is generally not convenient for

short data series since it assumes signals to be limited in both time and frequency (Ozaktas et al.,

1996). Algorithms for implementing the DFT such as The Fast Fourier Transform (FFT) are equally

inappropriate, as these do not estimate DFT with high accuracy (Becker and Morrison, 1996). Unlike

the DFT and FFT, the LSSF, models short periodicities that contain periodic or systematic signals,

which may contain either random or systematic errors or both (Vanicek, 1971). The performance of

this filter in removing the interference effects in the gravimeter data is deduced from comparison of the

spectral series for the gravimeter drift contaminated with noise (figure 2a), the filtered gravimeter tides

(figure 2b) and the synthetic tide (Figure 2c). The outputs of the filter for figures 2b and 2c are similar.

Noise registrations, above the white noise threshold are minimized by application of the filter.

0

100

200

300

400

500

0 2 4 6 8 10 12Frequency (cpd)

Figure 2 a: Spectrum of gravimeter drift before application of the LSSF showing high frequency noise

registrations

0

100

200

300

400


Figure 2b: Spectrum of gravimeter drift after application of the LSSSF

0

100

200

300

400

500


Figure2c: Spectrum of the synthetic tide

3.0 DATA ANALYSIS AND DISCUSSIONS

A rough comparison of some of the properties of the gravimeter and synthetic tide series is carried out

in table 1 to test the assumption of linearity in the propagation of the two tides. The properties are the

periodicity, and the amplitude and phase evolution ratios in the two data series. Where the ratios are

evaluated for the ith crest to the preceding (ith – 1) crest. For the amplitudes these are given as

11

;!! i

i

i

i

HH

hh

(2)

Where h and H stand for the amplitudes of the gravimeter and synthetic tides respectively. And for the

phases as

11

;!! "

"

"

"

Hi

Hi

hi

hi (3)

Where Φh and ΦH are the phases for the gravimeter and synthetic tides respectively. With ΦH - Φh defined

as the phase shift and h /H, the amplification. (Amplitude increase of gravimeter series over the

synthetic series).

The amplitudes and phase evolution of the series have been evaluated using the highest and the lowest

points of the envelopes in the filtered data and the shift is assessed from the correlation. (Figure 3) Figure 3: Correlation between gravimeter and synthetic tides for different time

lags(hours)

-0,2

0

0,2

0,4

0,6

0,8

1

-2,5 -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 2,5

lag / hours

co

rre

lati

on

Figure 3: Correlation between gravimeter and synthetic tides for different time lags(hours)

Table 1: Comparison of period, amplitude and phase evolution ratios of the gravimeter and synthetic

tides.

Gravimetric tide Synthetic tide Discrepancy

Average period (hours) 1180 11.80 0

Average amplitude evolution ratios 0.980 1.000 0.02

Average phase evolution ratios 1.04 1.05 0.01

The amplitude and phase ratios for the two series are close from Table 1. Hence there is a marked

linearity in the evolution of the two tides. Based on this established linearity we determined the lag in

hours by calculating the cross correlation coefficients at different adjusted values of the time lag,

taking the point of maximum correlation to imply zero lag (based on the linearity test) between the

two tides (with the synthetic tide taken as reference), i.e. we correlated H (t) and h (t + lag). Figure 3

is a plot of the correlations calculated for different lags. Since the data points were sampled at 1-hour

intervals, a second-order polynomial (0.0676 t2 + 0.288t +0.7011), where t is the lag/ hours, which

fitted the correlation curve properly, enables a more precise determination of the points of maximum

correlation (at the turning point of the curve). This we found to be at a time lag of about 1.4 hours.

Indicating that the gravimeter tide propagates with a time lag of about 1.4 hours behind the synthetic

tide. This of course introduces a phase lag of - Ω t0 , (~ 430 ) for the gravimeter series with respect to

the synthetic tide, where Ω is the frequency (Ω = 2π/T; T = period in hours) and t0 the time lag. The

amplitude variations given by the residuals (h-H) are also remarkable. They are higher by over 50%

when the time lags in the tides are not corrected (Figure 4), than when they are adjusted for lag even

only slightly (figure 5). Figure 5 is a lag-adjustment of only 1 hour, since data was sampled at 1-hour

intervals. This is less than the optimum lag-shift by 0.4 hours, but shows the effect of the shift in

reducing the residual amplitudes.

-2000

-1500

-1000

-500

0

500

1000

1500

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73

time / hour

grav

imet

er r

eadi

ng /

nm

s-

2

synthetic tide gravimeter tide residual

Figure 4: Comparison of gravimeter record with the synthetic tide

-2000

-1500

-1000

-500

0

500

1000

1500

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73

time/ hours

grav

imet

er re

adin

g /

nm

s-2

synthetic tide gravimeter tide residual

Figure 5: The two time series corrected for time lag by 1 hour and the residuals

3.1 Comments on the Amplitude Disparity

The amplification in the gravimeter data with respect to the synthetic tide has an average value of

0.65. Examination of the spectra for the gravimeter and synthetic tides (fig 4 & 5) indicates that the

observed tide has more energy than the synthetic tide in all the frequencies represented. The

broadening in the tidal band in the gravimeter spectrum especially between 2 and 4 cycles/day may

imply at first sight that more wave groups are contained in the gravimeter series but this is most

unlikely. Rather a meteorological artifact, most likely temperature is a strong factor that increased the

gravimeter spectral response in the tidal band. A temperature-induced systematic response of the

spring is by far the closest periodic signal that can fit this behavior. It shows evidence of growth with

time.

3.2 Comments on the Phase Lag

Astatised spring gravimeters such as the LCR gravimeter are known to record appreciable amplitude

damping and instrumental phase shift between the gravity sensor and the recording unit (Melchior,

1983; Torge, 1989). For short-time (< 1 day) tidal spectrum, Torge (1989) has reported damping

factors of 0.995 for the semidiurnal (m2) tidal wave and 10 and more for the phase lag. This is much

lower than the value we observed. The observed maximum cross correlation at lag 1.04 hr indicates a

hysteresis response of the gravimeter. A temperature based systematic error component is therefore the

most likely explanation for this creeping behavior.

4.0 CONCLUSIONS

The drift patterns of an old LCR gravimeter have been studied at a location in Nigeria to ascertain the

reliability of applying a LSSF for reducing abnormal drifting. Our results show a phase lag of the

gravimeter record as compared to an appropriate synthetic data due to instrumental response to

temperature changes. These resulted in very spurious signals that may limit the use of the instrument

for high precision work and tidal studies. Since surveying and static gravity observations require

accuracies of ~ 1mGal, this study implies that a LaCoste Romberg gravimeter that looses its

temperature compensation may be used under controlled conditions and with appropriate temperature

modeling for monitoring static gravity. It is however apparent that such precautions are over

simplifications, when the instrument is to be applied for monitoring geodynamic phenomena requiring

higher precision (µGal). For this second use there would be need for instruments with better

temperature compensations, perhaps new ones. The method adopted here is a fast way of to test the

LCR gravimeter response prior to field deployment. It does not represent a study of tidal phenomena

owing to the short period of the time series used.

ACKNOWLEDGEMENTS

This work was concluded during the authors’ research visit at the Abdus Salam International Center for

Theoretical Physics, Trieste Italy. The authors are grateful to Professor D.E. Ajakaiye, formally of the

University of Jos, Nigeria for providing the LCR gravimeter used for this study.

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transactions on signal processing, Vol 44, No 8, pp 2073-2077.

Durcame B, Sun H P, d’Oreye N, Van Ruymbeke M, Menajara I, (1999) Interpretation of the tidal

residuals during the 11 July 1991 total solar eclipse. Journal of Geodesy, 73,2, 53-55, 119 – 2357.

Honkassalo T (1975) Report on the application of geodetic measurements in the determination of

recent crustal movements in Finland. Tectonophysics, 29, Netherlands.

Kiviniemi A (1974) High precision measurements for studying the secular variations of Gravity in

Finland. Publications of the Finnish Geodetic Institute, No 60, Helsinki.

Macleod W, Turner D, Wright E (1971) The geology of the Jos plateau, Geological Survey of Nigeria

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A gravimetric quasi-geoid evaluation in the Northern region of Algeria using EGM96 and GPS/Levelling

Mohamed Aissa MESLEM

Laboratory of geodesy

Department of Research and Development National Institute of Cartography and Remote Sensing

123, Rue de Tripoli, BP 430, Hussein Dey, Algiers, Algeria

Abstract The use of GPS for the estimation of orthometric heights in a given region, with the help of existing levelling data requires the determination of a local geoid and the bias between the local levelling and the one implicitly defined when the geoid is calculated which is generally based on the gravity anomalies data. The heights of new data can be collected swiftly without using the orthometric heights from levelling; it is what one calls commonly levelling by GPS. In this framework, the Least Squares Collocation method (LSC) has been used to evaluate the quality of the available GPS-Levelling data, to determine a gravimetric geoid in the North region of Algeria and to estimate the constant datum bias. The data used in the setting of this study are: The geopotential model EGM96, a total number of 2534 gravity anomalies, as well as 43 GPS points connected to the geodetic network levelling present on the whole North part of Algerian. Keywords: Least Squares Collocation, GPS-Levelling, gravimetric Data, local geoid. 1. Introduction. The use of the GPS for the determination of the orthometric heights in regions where the levelling exist requires the determination of a local geoid and the estimation of the height constant bias between the gravimetric datum and local levelling since the local geoid will have its own zero level, whereas the data of levelling can be local, or national. The bias constant is estimated while calculating the mean of discrepancies between the differences of the GPS-Levelling and geoid. However, these values can have some irregularities in the spatial distribution; we must therefore counterbalance these discrepancies while taking in account their spatial correlations, this problem can be handled by the use of the Least Squares Collocation method (LSC). In the present paper this work is described by the use of the available data in Northern Algeria spreads from 32° in 37° N in Latitude and -4° in 10° E in Longitude. The main problem in this region is the one of the quality of the gravity data used and of GPS-Levelling. It has been necessary to withdraw the erroneous gravity values and to make a selection of GPS-Levelling points after a comparison between the observed values and those predicted by gravity.

2. Gravity data and GPS-Levelling. In practice, when one determine a gravimetric geoid locally, one must represent the neighbourhood of the gravity field by the use of a geopotential model of superior degree and order, in our case of study the harmonic spherical EGM96 has been used. Its contribution must be subtracted from the local data and must be restored thereafter. During the realization of this work, the only gravity data that were available were those provided by the International Gravimetric Bureau (BGI). We withdrew from the data file the duplicated gravity values as well as those appeared doubtful. As Digital Elevation Model (DTM), we had used the GTOPO30, the thinnest DTM that was available, of 1kmx1km resolution. 43 GPS-Levelling points with accuracy of ±50 cm have been used; these points have the advantage to be well distributed on the zone of study, however after a comparison between the observed values and those predicted by gravity, some points gave a difference up to the doorstep fixed beforehand according to the accuracy of the GPS-Levelling and the density of points. These points have been rejected before redoing another comparison. The EGM96contribution has been subtracted from the local gravity data. The statistics of the differences are presented in the table1. Table 1. Statistics of the EGM96 contribution on the 2534 free air anomalies.

mGal Mean Std. Deviation Observations 28.77 30.18 EGM96 34.14 23.90 Differences -5.37 23.64

A substantial agreement between EGM96 and the local gravity data have been obtained.

Fig.1 Free air anomalies after subtraction of the EGM96 contribution. Units mGal.

The RTM terrain effect reduction has been computed using a detailed 30”x30” grid and the outer zone height grid 5’x5’ from the reference height grid 30’x30’. After the subtraction of the EGM96 contribution and RTM terrain effect we obtain the residual anomalies by the following relation:

RTMEGMfaRés gggg Δ−Δ−Δ=Δ 96 (1)

Table2. Statistics of the RTM subtraction from the reduced anomalies. Units mGal.

# Min Max Mean Std. Dev.

sidualgReΔ -112.66 102.32 0.79 22.48

3. Least Squares Collocation. The LSC solution is gotten under the following shape:

jjijjii

N

iii

xCxLLb

LPTbPT

⋅=⋅+=

⋅=

−−

=∑

11

1

)()),(cov()(

),),(cov()(

σ

(2)

Where T is the local approximation of the anomalous potential, are the observations and jx

ijσ are the errors of covariance. The covariance is represented by the expression that follows

in which the constant R (Radius of the Bjerhammar sphere), a and A are determined from the local gravity data.

)(cos

)4)(2)(1()(cos

),,cov(),cov(1

1

212

2

2 ψψσ

ψ

k

k

Kkk

kK

kEGM P

rrR

kkkAP

rrRa

rrQP+∞+

+=

+

=∑∑ ⎟⎟

⎠

⎞⎜⎜⎝

⎛′+−−

+⎟⎟⎠

⎞⎜⎜⎝

⎛′

⋅=

′= (3)

P and Q are two points between a spherical distance, and r, r' are the distances of the two points from the origin. Initially an empiric covariance function has been determined of the reduced gravity anomalies. The estimated values have been adapted in that case to a covariance model by an iterative adjustment with the 3 parameters R , a and A. The limit summation has been fixed to 250, the coefficients bigger than 250 didn't give reliable information in the region. The depth of the Bjerhammar sphere ( ) has been estimated to -3.736 km and the total variance of gravity anomalies to 603.36 mgal².

BRR −

The Figure2 shows the empirical and analytical covariance functions.

Figure 2. Covariance functions of reduced gravity anomalies. The empirical covariance function of the residual gravity anomalies has as well been determined and the empirical values have been adapted to a covariance model. The limit summation for this time has been fixed to 260, The depth of the Bjerhammar sphere ( ) has been estimated to -9.961 km and the total variance of gravity anomalies to 432.83 mgal².

BRR −

The figure 3 shows the empirical and analytical covariance functions.

Figure 3. Covariance functions of residual gravity anomalies. The 2534 reduced and residual gravity anomalies have been used therefore to determine an estimated of T, from which the estimates of the geoid heights on the 43 benchmarks of GPS-Levelling have been computed. The results of the differences between observed values and predicted are presented in the table3. Table 3. Statistics of the differences between predicted and observed values. Units m.

43 points Min Max Mean Std.Dev.

GPS-Levelling 32.48 49.82 44.53 3.93

Prediction by reduced gravity

32.24 50.60 44.09 4.12

Differences -1.40 3.05 0.44 1.10

T

able 4. Statistics of the differences between predicted and observed values. Units m.


GPS-Levelling 32.48 49.82 44.53 3.93

Prediction by residual gravity

33.09 50.68 44.66 4.07

Differences -2.54 2.93 -0.13 1.35

The observation must be rejected if:

21

)(312

pjijijT

pippgGPS CErrCCCNN−

Δ +−⋅≥− and to the fixed doorstep. ≥

This has given several suspected errors, six observations have been rejected, the results of differences between the retained observed values and predicted are presented in the tables 5 and 6.

Table 5. Statistics of the differences between predicted and observed values. Units m.


GPS-Levelling 32.48 49.82 44.41 4.13

Prediction by reduced gravity

32.24 50.60 44.27 4.35

Differences -1.40 1.47 0.14 0.85

Table 6. Statistics of the differences between predicted and observed values. Units m.


GPS-Levelling 32.48 49.82 44.41 4.13

Prediction by residual gravity

32.27 50.74 44.28 4.31

Differences -1.52 1.68 0.12 0.92

The residual geoid undulations were determined by LSC, where the required auto and cross-covariance functions are computed by covariance propagation from the modelled local covariance function. The residual quasi-geoid is represented in the figure 4.

Figure 4. Residual quasi-geoid. Units m.

After restoring the long and short wavelength signals we obtain the final quasi-geoid presented below in the figure 5.

RTMEGMc NNNN ++= 96 (4)

Figure 5. Final quasi-geoid. Units m. 4. Constant bias. The parameters of which depend the used data, as the difference between the datum of the geoid and the local levelling, can be determined by LSC. The observations are tied to T and the vector of the X parameters by the following equation:

kKkk XATLx ε++= )( (5)

Where Lk is associated to the observation, Ak is a vector with the elements 0 or 1, X is the vector parameter and kε is the error of observation.

So :

iTT xCAACAX 111 )( −−− ⋅= (6)

The constant bias between the gravimetric datum and the local levelling has been estimated, using residual data. The results are presented in the tables below. Table 7. Statistics of differences between the residual gravimetric quasi-geoid undulations and GPS-levelling at 37 control points (in meters) before fitting.

Differences ζ rtm observed – ζ rtm predicted (m) Before fitting

Mean -.382 Std.Dev 1.079

Table 8. The parameter transformation model.

Data Parameter Units : m Values Error estimates

No Constant bias .128 .108 Table 9. statistics of differences between the residual gravimetric quasi-geoid undulations and GPS-levelling at 37 control points (in meters) after fitting.

Differences ζrtm observed – ζ rtm predicted (m) After fiting

Mean .000 Std.Dev .244

4. Conclusion. The gravimetric geoid has been computed by the Least Squares Collocation method. The subtraction of EGM96 gave the expected results, the variance and the mean value decreased significantly. The gravity RTM subtraction didn't reduce the variance a lot on the other hand the mean value has been reduced. This is due probably to the quality of the DTM used. The expected errors of the GPS-Levelling data are (±0.5 m), due mainly to the errors in the levelling. However, after the EGM96 subtraction and prediction by gravity, large differences (around 3 m) have been obtained between observed and predicted values for six stations. It might be antenna height problems or erroneous identification of the levelling points. It might also be due to tectonic movements in the period between the levelling and the GPS. Through this study we could see that the Least Squares Collocation method offers an important alternative to achieve an optimal evaluation in a stochastic process and to detect subsequently the gross errors of gravity data and GPS-Levelling. In addition and at last, the constant bias No between the gravimetric datum and the local levelling has been estimated to (0.128 m) with error estimate of (0.108 m). The local gravimetric geoid determined in the setting of this study deserves to be improved, by a densification of the gravimetric cover, the use of a more precise DTM and GPS-Levelling data with better quality.

5. References. [1] C.C. Tscherning, Geophysical Institute, University of Copenhagen, Haraldsgage 6, DK-2200 Copenhagen N, July 1990. The Use of Optimal Estimation for Gross-error Detection in Databases of Spatially Correlated Data. [2] C.C.Tscherning, Geophysical Institute, University of Copenhagen, Haraldsgage 6, DK-2200 Copenhagen N. Mathematical and statistical methods in physical geodesy, Spring semester 1991. Revised July 1992. [3] C.C. Tscherning. The Ohio State University, Department of Geodetic Science, July 1974. A FORTRAN IV Program for the Determination of the Anomalous Potential Using Stepwise Least Squares Collocation. [4] C.C.Tscherning, Department of Geophysics, Juliane Maries Vej 30, DK-2100 Copenhagen Ø, Denmark. Datum-shift, error-estimation for geoid determination, prepared for the International School on the Determination and Use of the Geoid. Draft, July 2002. [5] C.C.Tscherning (Department of Geophysics, Juliane Maries Vej 30, DK-2100 Copenhagen Ø, Denmark ) , Anwar Radwan, A.A.Tealeb, S.M.Mahmoud, Abd El-monum Mohamed, Ramdan Hassan, El-Syaed Issawy and K. Saker (all at National Research Institute of Astronomy and Geophysics, Helwan, Cairo, Egypt). Local geoid determination combining gravity disturbances and GPS/levelling A case study in the Lake Naser area, Aswan, Egypt. [6] Gravsoft – A System for Geodetic Gravity Field Modelling. C.C. Tscherning, Department of Geophysics, Juliane Maries Vej 30, DK-2100 Copenhagen N. R. Forsberg and P. Knudsen, Kort og Matrikelstyrelsen, Rentemestervej 8, DK-2400 Copenhagen NV. [7] S.A. Ben Ahmed Daho, S. Kalouche. Centre National des Techniques Spatiales, Arzew-31200, Algérie. Validation des Mesures Gravimétriques Par la Méthode de Collocation. Revue International des Technologies Avancées, janvier 2003. [8] H. Moritz, 1980. Advanced Physical Geodesy, H. Press, Karlsruhe-Tundridge Wells. [9] The Generic Mapping Tools (GMT), Paul Wessel, School of Ocean and Earth Science and Technology, University Of Hawaii at Manoa, and Walter H. F. Smith, Laboratory for Satellite Altimetry, NOAA/NESDIS/NODC, March 2002. [10] Lecture Notes, International School for the Determination and Use of the Geoid, Milan Oct. 1994, and Rio de Janeiro 1997. [11] M.A. Meslem, Résultats préliminaires relatifs à la détermination d’un géoïde du Nord de l’Algérie, Bulletin des Sciences Géographiques N.12, INCT, Oct.2003.

Improvement of the gravimetric model of quasigeoid in Slovakia Marcel Mojzeš, Juraj Janák, Juraj Papčo Department of Theoretical Geodesy, Slovak University of Technology in Bratislava, Slovak Republic Abstract. The paper presents two recent approaches of the quasigeoid modelling GMSQ03B and GMSQ03C in the area of Slovakia and their modifications after fitting by GPS/levelling method. The description of data sources follows the two different computational schemes. The statistical testing and fitting using 59 GPS/levelling points are presented. A brief discussion, conclusion and future perspective are summarized in the paper. Keywords. Gravimetric quasigeoid, data sources, GPS/levelling

1 Introduction Slovakia is predominantly mountainous country located in central Europe. There is the first part of the Carpathian belt. Slovakia had been systematically measured by detailed gravity measurements from 1956 to 1992 (Kubeš et al., 2001). All area of Slovakia had been covered by gravity observations with approximately homogeneous density varying from 3 to 6 points per square km, that represents more than 200, 000 gravity points. On this basis, using the generalized Molodensky’s theory, we have been trying for last nine years to compile the most detailed and accurate model of the quasigeoid. The first attempt was done in 1995 followed by the versions 1996, 1998A and 1998B. Especially the version 1998B after fitting into national vertical datum, known as GMSQ98BF – Gravimetric Model of Slovak Quasigeoid 1998 (Mojzeš and Janák, 1998) and (Mojzeš and Janák, 1999), has frequently been used for various scientific and practical applications.

Since 1998 the gravity database has been revised as far as blunders and systematic errors are concerned (Kubeš et al., 2001). Moreover the wider area of the mean gravity data has meanwhile become available. Some progress in theory and technology of computation has also been made, especially in combining of the terrestrial gravity data with the global geopotential model. All these circumstances have led us to our two recent approaches of the quasigeoid model in Slovakia: GMSQ03B and

GMSQ03C and their modifications after fitting GMSQ03BF and GMSQ03CF respectively. 2 Data sources Three types of input data were used for computation: Terrestrial gravity data (point and mean), elevation data (detailed and global) and global geopotential models. One paragraph is dedicated to describe the sources of a particular data type. As far as the location of all data is concerned, the reference system ETRS89 was used. Normal gravity field applied in computation was GRS80.

We used two sources of gravity data: point refined Bouguer gravity anomalies (232, 280 values) mainly within the Slovakia and mean values of the refined Bouguer gravity anomalies with resolution of 5′×7.5′ in the area 44° < ϕ < 56° and 12° < λ < 30°. The first source comes from detailed gravity measurements 1956-1992 mentioned in introduction. Both data sets were transformed to the gravity system GrS-95 (Klobušiak and Pecár, 2004) based on 16 absolute gravity points.

In order to transform the refined Bouguer gravity anomalies into Faye gravity anomalies, the elevation data were needed. We used two digital terrain models (DTM): national DTM within Slovakia and global DTM elsewhere. The first DTM is known as DMR2/ETRS89 and its resolution is 3″ in latitude and 5″ in longitude (Mojzeš, 2002). This resolution approximately corresponds to distance 100 by 100 metres. The vertical datum of mentioned DTM is Kronstadt, Baltic Sea with the abbreviation Bpv. The global DTM we used is well known GTOPO30 (http://edcdaac .usgs.gov/gtopo30.gtopo30.asp). This model was used outside of Slovakia. The unsolved problem we still have with the elevation data is the optimal connection of both national and global DTMs.

The last data type used in our solutions was the global geopotential model. We used two different models. First, the satellite only model GGM01 coming from GRACE dedicated satellite mission in tide-free system (Rummel et al., 2002). The second was well known

combined model EGM96 (Lemoine et al., 1998). 3 Computation schemes Both solutions GMSQ03B and GMSQ03C were in principle computed as so-called gradient solution of the linear Molodensky’s problem. Second term of the Molodensky series was approximated by terrain correction as recommended it by Moritz (1980). Correction of the part of the error coming from this approximation was applied in both solutions. The major differences between two solutions are in the manner of how the geopotential model is combined with the terrestrial data and how the integration of the residual gravity anomalies is performed. The first step, similar in both solutions, was the compilation of the residual Faye anomalies in a regular geographical grid Δϕ=20″ and Δλ=30″. First the refined Bouguer gravity anomalies had to be interpolated into mentioned grid. This was done using the Kriging interpolation technique, assuming the anisotropy in geographical coordinates. Then the interpolated refined Bouguer anomalies were transformed into Faye anomalies (free-air plus terrain correction) according to formula

HGgg RBF ρπ2+Δ=Δ (1)

where ΔgRB are the refined Bouguer gravity anomalies, G is the Newton gravitational constant, ρ is the density of topographical masses (we assumed constant value ρ=2670 kg/m3) and H stands for normal height in particular grid nodes obtained from DTM. Then the long wavelength part of the free-air gravity anomaly, assuming it approximately identical with the long wavelength part of the Faye anomaly, was subtracted from the Faye anomaly in order to obtain the residual Faye anomaly. The long wavelength parts for particular solutions were computed from global geopotential models GGM01 or EGM96 up to degree and order 20 or 360.

Solution GMSQ03B was computed using a low degree (n=20) spheroid obtained from the geopotential model GGM01. The aim was to avoid the errors coming from the higher-degree geopotential coefficients. Residual Faye anomalies were integrated using modified spheroidal Stokes’s function. Modification according to idea of Molodensky (1962) up to degree 20 was used. Integration of the residual Faye anomalies was performed using classical numerical integration up to spherical distance

ψ=3°. The truncation bias was estimated by EGM96 geopotential model using coefficients from 21 to 360.

Solution GMSQ03C was computed using a high degree (n=360) spheroid obtained from the geopotential model EGM96. Residual Faye anomalies were integrated using Spherical Stokes’s function. Applied integration technique was the Fast Fourier Transformation over the spherical rectangle integration domain identical with the area of input gravity data described above. The GRAVSOFT software package (Tcherning et al., 1992) was used for computation of residual part of the quasigeoid. The truncation bias was neglected.

Let us to describe both solutions mathematically and graphically. The quasigeoid model GMSQ03B consists of four terms

MoritztbresGGMBGMSQ ζζζζζ +++= 0103 (2)

where the first term represents the reference spheroid, the second term is residual part of the quasigeoid obtained from numerical integration of residual Faye anomalies, the third term is estimation of the truncation bias correction and the last term is correction of approximation of the second term of Molodensky’s series

21PPMoritz HG −= ργπζ (3)

computed according to Moritz (1980, Eq.(48-29)). In Eq. (3) the symbols G and ρ are obvious and γP is normal gravity and HP stands for normal height at the point of computation.

The quasigeoid model GMSQ03C consists of three terms

MoritzresEGMCGMSQ ζζζζ ++= 9603 (4)

where the meaning of particular terms is analogous to Eq.(2). Of course the first two terms of the right hand side were computed from the different data and the second term using even different integration technique. The third term in Eq.(4) computed from Eq.(3) is identical with the last term of Eq.(2).

The final quasigeod model GMSQ03B is shown in Fig.1.

17 17.5 18 18.5 19 19.5 20 20.5 21 21.5 22 22.5

Longitude (°)

48

48.5

49

49.5La

titud

e (°

)

33

35

37

39

41

43

45

47(m)

Fig.1 Quasigeoid model GMSQ03B Another model is not plotted, because the figure would be very similar, rather the difference GMSQ03B-GMSQ03C between both solutions is depicted in Fig.2.

17 17.5 18 18.5 19 19.5 20 20.5 21 21.5 22 22.5

Longitude (°)

48

48.5

49

49.5

Lat

itude

(°)

-0.20-0.14-0.08-0.020.040.100.160.220.280.340.400.460.520.580.640.700.76

(m)

Fig.2 Difference between GMSQ03B and GMSQ03C Both models are stored as a digital raster in geographical grid with the resolution of 20″×30″. 4 Testing and fitting As the models of geoid or quasigeoid become more accurate because of the advanced theory and also the quality and quantity of the data, the choice of the verification method and quality of the testing points become very important task. Therefore we chose the independent verification method GPS and levelling. Within the area of Slovakia we chose the 59 testing points shown in Fig.3.

17 17.5 18 18.5 19 19.5 20 20.5 21 21.5 22 22.5

Longitude (°)

48

48.5

49

49.5

Latit

ude

(°)

Fig.3 Distribution of the testing points These points belong to either Central European Geodynamic Reference Network (CEGRN), see (Hefty and Gerhátová, 1997), or Slovak

Geodynamic Reference Network (SGRN), see (Hefty, 1996). According to renowned estimates, e.g. (Stangl, 1998) and (Hefty and Mojzeš, 1999), the accuracy of the ellipsoidal height at these points in certain epoch is not worse then 2 centimetres. The accuracy of the normal heights determined by levelling over the Slovakia, in a relative sense, is also about 2 centimetres (ibid.).

The heights of quasigeoid models above the reference ellipsoid GRS80 were interpolated at the location of testing points and compared with the reference values obtained from the simple formula

Hhref −=ζ (5)

where h is the ellipsoidal height and H is the normal height according to Molodensky. The basic statistics of the differences is presented in Tab.1. Table 1 Basic statistics of the set of 59 differences Quantity ζGMSQ03B - ζref ζGMSQ03C - ζref Mean 0.334 m 0.711 m St. dev. 0.190 m 0.076 m Range 0.856 m 0.363 m

In order to use the quasigeoid model in

geodetic practice, e.g. for estimation of normal heights from GPS measurements, the adaptation of quasigeoid into national vertical datum, so called fitting, is necessary. Incompatibility of the gravimetric models with the national vertical datum is mainly due to influence of global geopotential models in gravimetric solutions, but the differences reflect also other influences and also errors. The fitting is always a “dangerous” process where the original quasigeoid model can be deformed and getting worse, especially between the fitting points. Therefore such a process should be treated with a special care. The best case would be to use the constant shift only to unify the vertical datum. If this is not possible, because the differences show some clear long wavelength features, the fitting surface of lowest possible degree should be used.

For fitting process we used the surface polynomial regression and the fitting points were identical with the testing points (Fig.3). The differences CBGMSQref /03ζζ − were

modelled as follows

( ) ( )( )qpP

p

Q

qqpQ ϕλλϕϕζ cos00

0 0, −−=Δ ∑∑

= =

(6)

In Eq. (6) the ϕ, λ are the ellipsoidal coordinates of fitting points, ϕ0, λ0 are chosen fixed coordinates and Qp,q are unknown coefficients estimated using the least squares method. The optimal degree of polynomial surface was chosen according to (Anděl, 1998)

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

4

2 1n

kA kk σ (7)

where Ak is the criterion, n is the number of fitting points, k is the degree of polynomial surface and σk is the standard deviation of residuals. The optimal degree is the lowest one where the stop criterion Ak decreasing. For both models of quasigeoid in our case, the second degree polynomial surface with 6 coefficients was estimated as the optimal. The coefficient values are shown in Tab.2. Table 2 Values and standard deviations of the fitting polynomial surfaces Quasigeoid GMSQ03B GMSQ03C Coefficient Value St. dev. Value St. dev. Q0,0 (m) -0,2903 0,0078 -0,7394 0,0075 Q1,0 (m/°) 0,0512 0,0138 -0,0697 0,0122 Q0,1 (m/°) -0,1807 0,0051 -0,0445 0,0048 Q1,1 (m/°2) -0,0560 0,0186 0,0015 0,0168 Q2,0 (m/°2) -0,0500 0,0261 -0,0576 0,0253 Q0,2 (m/°2) -0,0660 0,0054 0,0325 0,0053

Shape of the polynomial surfaces is shown in Fig.4. and Fig.5.

Fig.4 Fitting polynomial surface for GMSQ03B

Fig.5 Fitting polynomial surface for GMSQ03C Names of the quasigeoid models after fitting are GMSQ03BF or GMSQ03CF respectively. The standard deviation of residuals computed at the testing points after fitting is 0.041m for

GMSQ03BF and 0.032m for GMSQ03CF. Reader is encouraged to compare these values with the corresponding standard deviations before fitting in Tab.1. It is also interesting to see the residuals obtained at the testing points after fitting process plotted as surface maps, see Fig.6. and Fig.7, or as histograms, see Fig.8. and Fig.9.

17 17.5 18 18.5 19 19.5 20 20.5 21 21.5 22 22.5

Longitude (°)

48

48.5

49

49.5

Latit

ude

(°)

-0.1

-0.05

0

0.05

0.1

0.15

(m)

Fig.6 Residuals of GMSQ03BF

17 17.5 18 18.5 19 19.5 20 20.5 21 21.5 22 22.5

Longitude (°)

48

48.5

49

49.5

Latit

ude

(°)

-0.1

-0.05

0

0.05

0.1

0.15

(m)

Fig.7 Residuals of GMSQ03CF

-0.15 -0.1 -0.05 0 0.05 0.1 0.150

5

10

15

Fig.8 Histogram of GMSQ03BF residuals, x-axis in meters

-0.15 -0.1 -0.05 0 0.05 0.1 0.150

5

10

15

Fig.9 Histogram of GMSQ03CF residuals, x-axis in meters

5 Conclusion and perspective To rich 1-centimeter accuracy of the quasigeoid model in an absolute sense in a mountainous country, as e.g. Slovakia, is very difficult task. One of the problems is that even the accuracy of reference values at the testing points is sometimes worse than one centimetre. Some improvements can still be done in theory and technology of computation. Presented quasigeoid models can guarantee, as you can see from the results, accuracy better than 5 centimetres in an absolute sense. Of course in many local areas it is much better, also below one centimetre.

Our future investigation, in order to improve the accuracy of the quasigeoid model, will be oriented to rigorous computation of the second term of Molodensky’s series, refinement of digital terrain model, improvement of quality of the testing points, optimal numerical integration and careful unification of all reference systems. Acknowledgement The project of compilation of new quasigeoid model of Slovakia was partially supported by Research Institute of Geodesy and Cartography in Bratislava and partially supported by Grant Agency of the Slovak Republic (Project VEGA 1/1433/04).

For our computations we use several software products: commercial products SURFER and GRAPHER for interpolation of the input data and creating pictures, and scientific products SHGEO and GRAVSOFT software packages for working with global geopotential models and for numerical integration. We also use some, already mentioned, public data sets: GGM01, EGM96 and GTOPO30. References Anděl, J. (1998). Statistical Methods. Matfyz Press,

Prague. (in Czech) Hefty, J. and Ľ. Gerhátová (1997). Implementation of

BERNESE software 4.0, processing of connection measurements of selected AGS points with the SLOVGERENET, comparison of processing using the version 3.5 and 4.0 and it’s analysis. Technical report, Slovak University of Technology, Bratislava. (in Slovak)

Hefty, J. and M. Mojzeš (1999) Heights in Slovak

Geodynamic Reference Network. Proceedings of the workshop “GPS and heights”. TU Brno. (in Slovak)

Klobušiak, M. and J. Pecár (2004). Model and

algorithm of effective processing of gravity

measurements performed with a group of absolute and relative gravimeters. Geodetický a kartografický obzor 50 (92), No. 4-5, Prague, pp. 99-110. (in Slovak)

Kubeš, P., T. Grand, J. Šefara, R. Pašteka, M. Bielik

and S. Daniel (2001). Atlas of geophysical maps and profiles. Technical report. National geological institute, Bratislava. (in Slovak)

Lemoine, F.G. et al. (1998). The development of the

joint NASA GSFC and the NIMA geopotential model EGM96. NASA Technical Report NASA/TP-1998-206861, Goddard Space Flight Center, Greenbelt, Maryland.

Mojzeš, M. and J. Janák (1998). Gravimetric model of

Slovak quasigeoid. Proceedings of the Second Continental Workshop on the Geoid in Europe. Reports of the Finnish Geodetic Institute 98:4, pp. 277-280.

Mojzeš, M. and J. Janák (1999). New gravimetric

quasigeoid of Slovakia. Bolletino di Geofisica Teorica ed Applicata, Vol. 40, No. 3-4, pp. 211-217.

Mojzeš, M. (2002). Accuracy analysis of the digital

terrain model DMR-2 in ETRS-89 reference system. Proceedings of the VIII. International Polish, Czech and Slovak Geodetic Days. Polanica. Zdrój, pp. 45-51. (in Slovak)

Molodensky, M.S., V.F. Eremeev and M.I. Yurkina

(1962). Methods for Study of the External Gravitational Field and Figure of the Earth. Israel Program for Scientific Translations, Jerusalem.

Moritz, H. (1980). Advanced Physical Geodesy.

Abacus Press, Karlsruhe. Rummel, R., G. Balmino, J. Johannessen, P. Visser and

P. Woodworth (2002). Dedicated Gravity Field Missions – Principles and Aims. Journal of Geodynamics, Vol. 33, 1-2, pp. 3-20.

Stangl, G. (1998). The GPS Campaigns of CERGOP.

Reports on Geodesy, No. 9 (39), pp. 39-55. Tscherning, C.C., R. Forsberg and P. Knudsen (1992).

The GRAVSOFT Package for Geoid Determination. Proceedings of the First Continental Workshop on the Geoid in Europe. P. Holota and M. Vermeer (Eds.). Research Institute of Geodesy, Topography and Cartography, Prague, pp. 335-347.

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Satellite trace over the Earth

λ (°)

φ (° )

a = 6628 km

e = 0.002

i = 96.5 °

∆ T = 2 106 s

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Relative numerical precision:CLENSHAW*16 vs. FCR2*8

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Relative numerical precision:CLENSHAW*16 vs. PINES*8

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On the Potential of Wavelets for Filtering and Thresholding Airborne Gravity Data

M. El-Habiby, M. G. Sideris

Department of Geomatics Engineering,

The University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada, T2N 1N4

Abstract. Wavelets can be used in the decomposition and analysis of airborne gravity data. In this paper,

multiresolution analysis is applied to de-noise gravity disturbance and different de-noising techniques are

studied. The first objective is testing the usefulness of wavelets for analyzing and filtering airborne gravity

data. The second one is a comparison between the usage of the wavelet transform and other well known low-

pass filters. The gravity disturbances are filtered using wavelet thresholding to remove the noise introduced by

the dynamics of the aircraft. Two procedures have been tested. The first one is de-noising using minimax and

universal techniques. The second one is a combination of wavelet thresholding and filtering at certain levels.

Comparison to independent reference data is performed in the area of interest to determine the external

accuracy of this approach. Results from both cases and from the low-pass filter approach are compared. The

results of testing different de-noising techniques show that the combination of thresholding and filtering can

reach RMS values equal to 25 mGal in comparison to the results from the 90s low-pass filter.

Keywords. Wavelet multiresolution analysis; airborne gravimetry; filtering; universal, hard and soft

thresholding.

1 Introduction

Wavelet analysis is a comparatively young branch in signal processing. It is developed to overcome some of

the problems of Fourier analysis. Wavelet expansions allow better local description and decomposition of

signal characteristics. Filtering airborne gravimetry data has been introduced in a number of publications.

Glennie and Schwarz (1997), Schwarz and Glennie (1998), Glennie and Schwarz (1999), Wei and Schwarz

(1998), and Glennie (1999) low-pass filtered gravity disturbances using an FIR filter to specified cut-off filter

lengths, which range from 30s to 120s.

Bruton et al. (1999) used wavelets for the analysis and de-noising of kinematic geodetic measurements.

Wavelets have been used as a de-noising tool for removing stochastic noise from different geodetic signals.

They used SURE (Stein Unbiased Estimate of Risk) soft thresholding and tested the Daubechies family in the

decomposition of the signal. Abdel-Hamid et al. (2004) improved the performance of MEMS-based sensors

using multiresolution analysis. They concentrated on enhancing the performance of Kalman-filtering

INS/GPS integration techniques.

Our objective in this paper is to check the effectiveness of using wavelets in de-noising gravity disturbance.

Different thresholding techniques will be tested. Minimax thresholding, fixed thresholding (universal), and

thresholding combined with filtering at certain levels will be used. Hard and soft thresholding have been used

in parallel with the different thresholding techniques.

In the next section, wavelets are introduced as a filtering tool. Then the system, flight and data

characteristics are described. This is followed by the analysis of the data using wavelets and FFT. Following

that the results of de-noising the gravity disturbance data using different wavelet thresholding techniques are

compared with reference data and a comparison is made between the different thresholding and filtering

techniques. The paper ends with conclusions, recommendations and future work with respect to the

suitability, accuracy and efficiency of the methods used.

2 Wavelets as a Filtering Tool

A wavelet base is a set of building blocks to construct or represent a signal or function; Burrus et al. (1998).

The discrete wavelet transform (DWT) coefficients k,jω of a signal or a function )t(f are calculated by the

inner product

)t(),t(f k,jk,j ψω = (1)

where k,jψ is the wavelet expansion function, and both j is the scale and k is the translation and both are

integer indices. The inverse wavelet transform is used for the reconstruction of the signal from the wavelet

coefficients kj,ω by

=j k

k,jk,j )t()t(f ψω (2)

Equations (1) and (2) are named analysis and synthesis, respectively. Wavelets used in this paper have energy

concentrated in time, continuous null moments, and decrease quickly towards zero when the input tends to

infinity. Meyer and Daubechies wavelets have been used in this research (Figure 1). Both the wavelet and

scaling function for Meyer are defined in the frequency domain and have a closed form. Meyer is not

compactly supported, nevertheless, it introduces a good approximation leading to FIR filters, and then

allowing DWT, (Misiti et al., 2002). Daubechies (db) wavelets have no explicit expression except of db1

(Haar wavelet).

a b

Fig. 1 Daubechies 4 wavelet (a) and Meyer wavelet (b) (Misiti et al., 2002)

2.1 Wavelet Thresholding

Wavelet thresholding is a technique used to remove random noise and outliers from the signal before

reconstruction. Wavelet absolute coefficients larger than a certain specified threshold δ are the ones that

should be included in reconstruction. The reconstructed function can be show as, (Ogden, 1997):

)t(I)t(f k,jk,jj k k,j

ψωδω

=>

(3)

where δω >k,j

I is the indicator function of this set. This represents a keep or kill wavelet reconstruction. This

thresholding is a kind of nonlinear operator on the wavelet coefficients vector. This leads to a resultant vector

of estimated coefficients k,jω to be used in the reconstruction process. The problem is always about finding

the proper thresholding value. In order to make this decision, two main parameters have to be taken into

account. These parameters are the sample size n and the noise level 2σ . The estimated coefficients can be

determined from either hard thresholding or soft thresholding. In case of hard thresholding the estimated

coefficients will be:

≥

=otherwise0

ifˆ k,jk,j

k,j

δωωω (4)

In case of soft thresholding, the estimated coefficients will be:

−<+

≤

≥−

=

δωδω

δω

δωδω

ω

k,jk,j

k,j

k,jk,j

k,j

if

if0

if

ˆ (5)

This can also be illustrated in the following figure.

a b

Fig. 2 (a) Hard and (b) soft thresholding

Choosing the value of the threshold is a very fundamental problem in order to avoid oversmoothing or

undersmoothing.

Minimax thresholding has been applied in this research. This technique depends on the sample size n and it

minimizes a bound for the risk involved in estimating a function. Minimax threshold has no closed form, but

it can be approximated numerically. Another thresholding technique has been used in this research; it is called

universal thresholding (fixed thresholding). This is an alternative to the minimax technique. The thresholding

value is chosen equal to nlog2 . This thresholding value is larger than the corresponding value in the

minimax estimate, for the same sample size; for more details, see Donoho and Johnstone (1994).

2.2 Wavelet Level-dependent Thresholding and Filtering

Filtering is adjusting all the coefficients of certain level or levels to zero:

≥

≤=

max

maxk,j

k,jjjif0

jjifˆ

ωω (6)

In this research, filtering has been combined with thresholding for de-noising gravity disturbances. First,

either minimax or universal thresholding is applied to the signal. This is followed by applying filtering to a

number of the high wavelet decomposition levels. The choice of the wavelet decomposition level to be

filtered depends on the minimum wavelength of signal that can be reconstructed from each level. Sometimes

both filtering and level-dependent thresholding are used, which means that a possibly different threshold

value is chosen for each wavelet level j, (Barthelmes et al., 1994).

3 System, Flight and Data Description

The data used originated from a project collected by the University of Calgary on 9, 10, and 11 of September,

1996. Only the data of the second day has been used in this research. The hardware used consists of three

main components: a strapdown INS, GPS master and remote stations, and data acquisition. The INS system is

a Honeywell LASEREFIII (LRFIII) navigation grade inertial system. Various types of GPS receivers have

been used. The major requirements were low noise and high reliability. The data was collected over the

Rocky Mountains; the area covered was 100 km * 100 km. This area was covered by 14 lines in day 2, as

shown in Figure 3.

Fig. 3 Flight pattern for day 2 (September 10th) of the Kananaskis Campaign

This area was chosen because of the high variability of the gravity field and the availability of dense ground

gravity values to be used as reference. The reference data were upward continued gravity disturbances with

RMS equal to 1.5 mGal. The heights of the terrain in the area vary from 800 m to 3600 m. Average ellipsoidal

flight height was 4357 m. The flight speed was 360 km/hr. The data used in the de-noising procedure has been

resampled to 1 Hz (01 h bandlimit).

4 Analysis and De-noising of Results

4.1 Wavelet versus FFT analysis

The data to be analyzed is a gravity disturbance with 1h bandlimit, introduced as sub output from the

KINematic Geodetic Software for Position and Attitude Determination (KINGSPAD) software, figure 4.

Fig. 4 Gravity disturbance with 1 Hz sampling rate

The gravity disturbance analysis (Figure 5a) starts with the demonstration of (1h bandlimit) signal using

Fourier analysis. The FFT was used to visualize the different frequency contents of the signal. This spectrum

shows that there are different signals at different frequencies, but the problem is that it is difficult to localize

the errors, and separate them from the signal (figure 5a). The deficiency of time localization by FFT analysis

leads to the use of wavelet transform analysis.

(a)

(b)

Scale of colors from MIN to MAX

Ca,b Coefficients

Sca

le

1

5

9

13

17

21

25

29

33

37

41

45

49

53

57

61

(c)

Fig. 5 (a) FFT spectrum shows a number of undesired frequencies. (b) Time-frequency analysis and localization

using continuous wavelet transform. (c) Coefficients Line - k,jω for scale j = 32 (frequency = 0.031).

The usage of the continuous wavelet transform allows time-frequency localization; this can be seen in Figure

5b. Continuous here means that it is not decimated to dyadic format (decreasing the number of coefficients

into half at each approximation or detailed level) as in the traditional discrete cases. However, all the wavelet

decomposition coefficients are taken into account. Darker shades show high frequencies, different from the

expected gravity disturbances frequencies (depending on a priori information). These high frequencies are

interpreted as errors at certain time and scale. In wavelet analysis, different types of errors can be tracked

through the whole trajectory and interpreted corresponding to different aircraft dynamics. Also, by the

decomposition of the signal into several levels, stochastic errors and outliers can be easily detected and

removed using different thresholding techniques. For example, the takeoff and different maneuver periods

between lines can be easily identified. Figure 5c is a sample of the coefficients at scale 32 showing clearly the

maneuvers between lines.

4.2 Wavelet versus low-pass filter

Wavelet transforms have been also used in de-noising and smoothing of gravity disturbances. The wavelet

techniques are compared to the output of a low-pass filter with four cut-off lengths, which are 30s, 60s, 90s,

and 120s. The difference between output computed raw data and the reference data shows the presence of

noise and outliers, as shown in figure 6. The four low-pass filters are an output from the KINGSPAD and

AGFILT software, developed by the University of Calgary.

Fig. 6 Deference between reference and bandlimited 1h data

The difference between the output from the four low-pass filters and the reference data is shown in Figure 7.

The RMS of the difference between the low-pass filtered data and the reference has been computed.

(a)

(b)

Fig. 7 (a) Difference between reference data and 30s, and 60s low pass filter output. (b) Difference between reference data and 90s, and 120s low pass filter output.

The same data was de-noised using different thresholding and filtering techniques, using the two wavelet

families shown in Figure 1. Finally, they are compared with the results obtained from the four low-pass filters.

The universal and minimax thresholding has been applied to the gravity disturbance data. Both of them

have been applied with soft and hard thresholding. The results are compared to the reference data. The RMS

of the residuals ranges between 712 and 1700 mGal, which is not acceptable (Figure 8).

(a )

(b)

Fig. 8 (a) Difference between the gravity disturbances de-noised by universal (fixed) thresholding and the reference

data (b) Difference between the gravity disturbances de-noised by minimax thresholding and the reference data

Two modifications have been made to reduce this difference. Another wavelet family has been tested and

filtering has been applied to the data in sequence with the two thresholding techniques mentioned before.

After different trials and tests, it has been found that the usage of minimax soft thresholding, by Meyer

wavelets followed by filtering at certain high levels of the wavelet decomposition, is the optimum in our case

study.

In the following figures it can be seen that universal thresholding has been tested with Meyer wavelets. The

RMS was 729 mGal, which is still not acceptable. This is followed by three trials of applying the minimax

soft thresholding, and applying filtering by removing a whole decomposition level at a certain scale. The

value of the RMS decreased from 1846 mGal, because of inappropriate choice of the filtering level, at the first

trial to 25 mGal at the third trial. This value is close to the best result introduced by low-pass filtering (120s).

This can be illustrated in figure 9.

(a)

(b)

Fig. 9 (a) Meyer wavelets with fixed soft thresholding (up), and minimax and filtering (1st trial) down. (b) Meyer wavelets with minimax and filtering 2nd (up), and 3rd (down) trial

Table 1 RMS of the difference between de-noised gravity disturbances and reference data

Method RMS (mGal)

Low pass filter (30s) 99.52




Minimax & filtering 1 1846.80



The choice of the thresholding value is depending on the sampling rate and the signal that can be

reconstructed at each level. It is automated through one of the two techniques mentioned above, either

universal or minimax thresholding. Applying filtering to certain levels is very dangerous, needs a prior

information and is depending on trial and error. The same procedure was applied to one of the 14 lines, which

is line 6 between 4725 s and 5280 s. The first and last 10 points, which contain a lot of outliers because of the

dynamics introduced from the turning maneuvers of the aircraft, have been removed. Similar results were

obtained The only difference is that, in case of line 6, less usage of filtering was required because of the

stability of the aircraft during this period. It is worth mentioning here that a trend has been removed from the

difference of the de-noised and the referenced data, in both cases (low-pass filter, and different wavelet

procedures). The RMS errors from different trials and approaches for the difference between the de-noised

gravity disturbance and reference data are summarized in table 1.

5 Conclusions

Wavelets can be used in the decomposition and analysis of airborne gravimetry data. Minimax and fixed

thresholding techniques as an automated procedure are not enough for de-noising airborne gravity data,

because of the need of filtering high frequency levels. The combination of these automated thresholding

techniques with filtering is more effective in de-noising and bandlimiting the gravity disturbance. Soft

thresholding proved to be more effective in this study than hard thresholding. Different wavelets have

different effect on the analysis and de-noising of the signal. This can be easily noticed from the improvement

in the accuracy when using Meyer wavelets instead of Daubechies. Decomposition of the signal to higher

levels and applying thresholding and filtering is effective especially in the case of outliers. Wavelet de-noising

reached the same accuracy as the approximation introduced by the low-pass filters with 120s cut-off, and

better than the 30s, 60s, and 90s low pass filters. Trial and error is required for determining the levels to be

removed (filtering) in case of combining both filtering and thresholding. Further research is under

investigation to automate the filtering and thresholding procedures by combining them with priory

information about the range of the frequency required.

References

Abdel-Hamid, W., A. Osman, A. Noureldin, and N. El-Sheimy (2004). Improving the performance of

MEMS-based inertial sensors by removing short-term errors utilizing wavelet multi-resolution analysis.

Published in the ION National Technical Meeting, Ca, USA, 26-28 Jan.

Barthelmes, F., L. Ballani, and R. Klees (1994). On the Application of wavelets in Geodesy. In: "Geodetic

Theory Today", convened and ed. by Sansò, F., Springer-Verlag Berlin, Heidelberg 1995 (International

Association of Geodesy Symposia 114, L'Aquila, Italy, May 30 - June 3, pp. 394-403.

Bruton, A. M., K. P. Schwarz, and J. Skaloud (1999). The use of wavelets for the analysis and de-noising of

Kinematic geodetic measurements. Geodesy Beyond 2000-IAG General Assembly, Birmingham, UK, July

19-24.

Burrus, C. S., R. A. Gopinath, and H. Guo (1998). Introduction to Wavelets and Wavelet Transforms: A

Primer. Prentice Hall, Upper Saddle River, New Jersey, USA.

Donoho, D. L., and I. M. Johnstone (1994). Ideal Spatial Adaptation via Wavelet Shrinkage. Biometrika 81:

pp. 425-455.

Glennie, C. (1999). An analysis of an Airborne Gravity by Strapdown INS/GPS. Ph.D. Thesis, UCGE Report

#20125, Department of Geomatics Engineering, University of Calgary, Calgary, Canada.

Glennie, C., and K. P. Schwarz (1999). A Comparison and Analysis of Airborne Gravimetry Results from

Two Strapdown Inertial .DGPS Systems. Journal of Geodesy, 73, Issue 6, pp. 311-321.

Glennie, C., and K.P. Schwarz (1997). Airborne Gravity by Strapdown INS/DGPS in a 100 km by 100 km

Area of the Rocky Mountains. In: Proceedings of International Symposium on Kinematic Systems in

Geodesy, Geomatics and Navigation (KIS97), Banff Canada, June 3-6, pp. 619-624.

Misiti, M. Y., G. Oppenheim and J. Poggi (2002).Wavelet Toolbox for use with Matlab. User’s Guide, the

Math Works Inc.

Ogden, R. T. (1997). Essential Wavelets for Statistical Applications and Data Analysis. Birkhäuser, Boston,

USA.

Schwarz, K. P., and C. Glennie (1998). Improving Accuracy and Reliability of Airborne Gravimetry by

Multiple Sensor Configurations. In: proceeding of International Association of Geodesy Symposia

’Geodesy on the Move’. Forsberg R. Feissel M. Dietrich R. (eds) Vol. 119, Springer Berlin Heidelberg

New York, pp 11-17.

Wei, M., and K.P. Schwarz (1998). Flight Test Results from a Strapdown Airborne Gravity System. Journal

of Geodesy, 72, pp. 323-332.

1

A comparative study of the geoid-quasigeoid separation term C at

two different locations with different topographic distributionsMuhammad Sadiq A1, Zulfiqar Ahmad A2

Department of Earth Sciences, Quaid-i-Azam University, Islamabad Post Code 45320, PakistanA1: [email protected], A2: [email protected]

Abstract:

The present study has been carried out to compare the geoid-quasigeoid separation in two areas of

Pakistan with different topographic distributions. The height datum of Pakistan is based on the

orthometric height system. Since Pakistan has a diversity of terrain distribution as regards the

elevation from mean sea level due to vast expanse comprising both land and hilly areas, the emphasis

of this study has been placed on the quantification of the geoid-quasigeoid separation term with

respect to elevation distribution for future geoid determination. Bouguer gravity anomalies and digital

terrain elevation were used to estimate the minimum/maximum separation between these two

reference surfaces. This was also compared with a separation term C computed from EGM96 gravity

anomalies. The statistics of the results in the two areas exhibit a one to one correspondence of

EGM96 gravity anomalies with observed gravity data and digital elevation data. The area with high

mountains (Kalat) has more offset between the two surfaces. The standard deviation of separation

term is ~77 mm from observed and 62.5 mm from model data. In contrast with the low elevation area

(Dadu), the standard deviation of the separation becomes as small as ~2 mm from observed data and

~7mm from gravity anomalies computed from the EGM96 model. The terrain correction has

measurable effect on the standard deviation in Kalat and is very insignificant in Dadu area. The

difference of the separation term from observed data and the model can be related to assumption of

the topographic density, local mass irregularities and inherent omission error of the EGM96 model.

These results also show that the separation term C is significant in Pakistan and may be required to be

incorporated in the final geoidal solution.

Keywords: Geoid, Quasigeoid, Bouguer Gravity anomaly, Height Anomaly, C Term

1-Introduction:

The investigation made in this paper is a comparative study of the effect of terrain on the geoid to

quasigeoid separation in two selected areas of Pakistan. This study also focuses on the issue of

significance of this term with reference to Pakistan and surrounding areas for onward more feasible

geodetic boundary values solution.

The geoid is an equipotential surface of the Earth that corresponds to mean sea level, whereas the

quasi geoid is a geometrical surface referred to a normal height system. The geoid undulation (N) is

the separation between the ellipsoid and the geoid measured along the ellipsoidal normal. The height

anomaly (ζ ) is the separation between the reference ellipsoid and quasigeoid along the ellipsoid

normal. There is similar concept of orthometric heights (Ho) measured along plumb line whereas,

mailto:[email protected]

mailto:[email protected]

2

normal heights (HN) are measured along the ellipsoidal normal. These reference surfaces of the

geodesy are shown in Figure 1

g(rt ,Ω )

Topography

ζ

ζ

Telluroid

HN Ho

Wo Quasigeoid

N

Uo

Figure-1: The geoid undulation(N), Orthometric heights (H), Height Anomaly (ζ ) and Normal

heights (HN)

In this study, a comparison is made of the geoid to quasigeoid correction term for two study areas

with different topographic distribution. This comparison is made with observed gravity/elevation

data, gravity anomalies computed from the EGM96 global model and digital elevation model data

extracted from Shuttle Radar Topographic Mission (SRTM) for these areas. The EGM96 global

model is a recent estimate of the global gravity and height anomalies [Rapp (1997)]. The global

correction term was determined using EGM96 potential coefficients. The Bouguer gravity anomalies

and elevation data of the 3495 points in the low land area (Dadu) and 927 points in the high elevation

(Kalat) area were determined from observed land gravity data while orthometric heights were

determined using spirit leveling. The terrain correction was also calculated in these areas and added

to the Bouguer anomaly to quantify the effect of terrain on geoid to quasigeoid separation.

2-Theoretical Background

Molodensky et al. (1962) formulated the geodetic boundary value problem on the earth surface and

introduced two new surfaces called the telluroid and the quasigeoid. The quasigeoid is not an

equipotential surface of the earth’s gravity field and thus has no physical meaning (Heiskanen and

Moritz, 1967). However, the quasigeoid can be determined somewhat more directly from surface

gravity data without prior knowledge of the topographic bulk density. The separation between the

reference ellipsoid and the quasigeoid is called the height anomaly and is defined as;

Geoid

Ellipsoid

3

ζ = h - HN (1)

The telluroid is the surface defined by plotting the points at a distance equal to ζ below the earth

surface. The distance between the ellipsoid and the telluroid is called the normal height HN and can

be computed from optical leveling measurements using the integral mean of normal gravity between

the reference ellipsoid and the telluroid as,

γ = dHH

1NH

0N ∫ γ (2)

This equation makes it possible to determine the normal heights without prior knowledge of the

topographic density distribution along the ellipsoidal normal.

The orthometric height Ho of any point on the surface of the earth is the height of point above the

geoid along the geoidal normal. It can be determined by optical leveling measurement along the

plumb line between the point on surface and the geoid. It requires the integral mean of the gravity

along the geoidal normal and can be determined using the relationship [Heiskanen and Moritz

(1967)].

g = dHgH1 oH

0o ∫ (3)

However, this requires the knowledge of the subsurface density along the plumbline and this

information is not available in normal routine work. For this purpose, the Poincare-Prey gravity

gradient is often used. Assuming that the ellipsoidal normal and plumb line are coincident between

the earth surface and the geoid, the geoid separation can be approximated by

N ≅ h - Ho (4)

Eliminating h from Eq. 1 & 4 gives

HN – Ho = N - ζ ≅ C2 (5)

Where, C2 is the geoid-quasigeoid separation term. It depends on the Bouguer anomaly, average

theoretical gravity and orthometric height of the point. This term can be derived as follows.

The basic formula for the definition of orthometric height is given by (Heiskanen and Moritz, 1967)

Ho =g

rC t )]([ Ω )20;2/2/(),( 0 πλπϕπλϕ ≤≤≤≤−Ω∈Ω∀ (6)

Where C[rt (Ω )] is the geopotential number, and g (Ω ) is the mean value of the gravity along the

plumb line between geoid and earth surface determined using Poincare-Prey’s gravity gradient.

The Molodensky’s normal height HN (Ω ) reads (Molodensky, 1945)

HN = γ

)]([ ΩtrC ΩΩ ∈∀ o (7)

4

Where γ (Ω ) is the mean value of normal gravity along the ellipsoidal normal between the surface

of the geocentric reference ellipsoid and the telluroid.

The difference between the normal height and orthometric height can be determined by the following

relation.

HN – Ho = Ho(Ω ) )(

)()(gΩγ

ΩγΩ − ΩΩ ∈∀ o (8)

The difference between the mean gravity g (Ω ) and mean normal gravity γ (Ω ) can be

determined using their mathematical definitions with some assumptions as. [Heiskanen and Moritz,

1967]

g (Ω - γ (Ω ) = g [rt (Ω )] - γ [ro (Ω )+HN] - 2 )(HG oo Ωρπ (9)

Where G is Newton’s gravitational constant and oρ is the mean topographical density oρ = 2.67

g.cm-3. The expression on the right side of Eq. 9 is the Simple Bouguer Anomaly. Therefore Eq. 8

can be written for geoid – quasigeoid separation term as,

HN – Ho = )(

gB

Ωγ∆

H (Ω ) (10a)

or

pN - pζ = )(

gB

Ωγ∆

H (Ω ) (10b)

Eq. 10 is also called the C2 term as mentioned by Sjoberg, 1995, Rapp, 1997 and Featherstone and

J.F. Kirby, 1998.

),(2C),(1C)r,,(),(N Eo λϕλϕλϕζλϕ ++= (11)

or

),(2C),(1C)r,,(),(N Eo λϕλϕλϕζλϕ +=− (12)

where

Hh

Hr

),(1C∂∂

∂∂

+∂∂

=γ

γζζ

λϕ (13)

and

C2 = )(

gB

Ωγ∆

Ho(Ω ) (14)

However, the Sjoberg (1995) paper includes a term dependent on H2 as well as H in the N - ζ

difference. For this analysis, only the 1st order term in terrain elevation (Ho) is considered.

5

Here Bg∆ is the Bouguer gravity anomaly, γ (Ω ) is the average normal gravity between the

reference and the telluroid. It is calculated between geoid and earth surface in routine work taking

orthometric height as height term in its calculations.

3-Data Analysis and numerical comparison of the Results:

The data in both study areas comprise the absolute gravity along with orthometric heights. In addition

to this Digital terrain model data (3 arc sec.≈ 90 m) from SRTM (USGS, EROS data center) was used

to compare its results with orthometric heights to asses its applicability in the terrain correction

determination for future geoid computation in Pakistan. The area boundaries and elevation statistics

are shown in table 1. The Bg∆ was computed using generalized formula [W.E. Featherstone and

M.C. Dentith (1997), Eq-23] using point absolute gravity and elevation/DTM data. Theoretical

normal gravityγ was computed using Somigliana’s formula. For the computation of Bg∆ , standard

topographical density was assumed, i.e. 2.67 g.cm-3. The Bouguer gravity anomalies and average

theoretical gravity in conjunction with elevation data at the corresponding points were used in Eq. 14

to compute the C2 correction term. The results from absolute gravity and terrain/DTM data were

compared with the geoid to quasigeoid correction term determined for the EGM96 global

geopotential model [Lemoine et al. (1997), National Imagery and Mapping Agency]. The contour

plots of C2 term from observed data is shown in figure 6 & 7. The statistics of the C2 term results

are shown in table 2.

The Free Air gravity anomalies are correlated (90%) with elevation with inverse correlation (62%) of

Bouguer anomalies in Kalat area (Fig-2). The correlation of Bouguer anomalies indicates the

considerable Bouguer plate effect demanding the application of terrain correction in this area. This

correlation is not significant (Fig-3) in Dadu, as both anomalies are negative and behave

approximately similarly with elevation. This small difference of free air and Bouguer anomalies can

be related to a small Bouguer plate effect (1.5 to 15.5 mgal only).

R2FAA = 0.90

R2BA = 0.62

-200-150-100-50

050

100150200

0 500 1000 1500 2000 2500 3000

Elevation(m)

Fre

e A

ir an

d B

ougu

er G

raA

nom

alie

s(m

gal)

Fig-2. Relationship of Elevation with Free Air and Bouguer gravity Anomalies in Kalat Area

6

R2BA= 0.28

R2FA= 0.16

-120

-100

-80

-60

-40

-20

0

0 50 100 150

Elevation(m)

Fre

e A

ir a

nd B

ouguer

Anom

alie

s (

mgal)

Free Air AnomalyBouguer gravity anomaly

The comparison of geoid to quasigeoid separation terms from observed gravity data and gravity

anomalies derived from the EGM96 global geopotential model in Kalat area shows close correlation

with each other. The use of the digital terrain model in Kalat has created insignificant difference to

the computed C2 term. Although DTM data differs much (of the order of 10-40 m), the effect on the

resulting separation term is insignificant due to the reason that the coefficient of H in Eq-14 is of the

order of 10-3. The maximum magnitude of the C2 term determined from observed gravity data is 440

mm while that of the EGM96 gravity anomalies is 484mm. However, the standard deviation differs

only by 14.3 mm. This term from gravity measurements in Dadu area comes out to be 12 mm while

19.6 mm from EGM96 gravity anomalies whereas, standard deviation differs by only 5mm. The

results of DTM data confirm the observed data. This indicates similarity of the elevation data from

DTM with leveling data. The difference of values from observed gravity data and EGM96 model may

be related to the omission error within EGM96 model. This shows that the overall effect of omission

error in Dadu is not significant, whereas it is considerable in Kalat.

The terrain correction was calculated using the software developed by Yecai Li, and Michael G.

Sideris (1993) on the grids of orthometric heights to study its effect on this term. The terrain

correction is not very large however considerable in Kalat and it has practically insignificant effect on

this term in Dadu area. The standard deviation of geoid-quasigeoid separation differs by only 5.4 mm

in the Kalat area and really no effect in Dadu area.

Table-1:Statistics of altitude (in meters) of 927 land gravity points in Kalat(high land) area and 3495

points in Dadu (low land) area along with Terrain correction and latitudinal/longitudinal boundaries.

Measured

Parameter(Kalat area)

Max Min Mean Measured

Parameter (Dadu Area)

Max Min Mean

Longitude(degrees) 67.093 66.496 66.776 Longitude(degrees) 67.78 67.35 67.566

Latitude(degrees) 29.426 28.57 29.005 Latitude(degrees) 27.29 26.53 26.909

Altitude above MSL(m) 2782.8 1006.2 2024.7 Altitude above MSL(m) 136.5 13.7 34.008

Altitude (from DTM) 2782 1021 2029 Altitude (from DTM) 148 32 51

Terr. Correction (mgal) 7.663 0.034 0.819 Terr. Correction(mgal) 0.155 0.0 0.005

Fig-3. Relationship of Elevation with Free Air and Bouguer gravity Anomalies in Dadu Area

7

Description of

Model(Kalat Area)

Max Min Mean Std.

Dev.

Description of

Model(Dadu Area)

Max Min Mean Std.

Dev.

C (Obs. Gravity )

Before Terrain

Correction applied

-103 -440 -297 76.8 C2(Abs. Gravity )

Before Terrain

Correction applied

-0.7 -12 -2.67 1.6

C (Obs. Gravity)

After Terrain

Correction applied

-106.9 -427.4 -260 71.4 C2(Obs. Gravity)

After Terrain

Correction applied

-0.66 -10.1 -2.3 1.46

C2 (DTM) -103 -437.4 -295.7 76.6 C2 (DTM) -1.1 - 10 - 3.2 1.5

C2(EGM96) -210 -484 -411.5 62.5 C2(EGM96) 18 -19.6 0.7 7

Table-2: Statistics of the geoid to quasigeoid correction term C2 (in mm) at 927 land gravity points in

Kalat (high land) area and 3494 points in Dadu(low land) area from the Bouguer gravity anomaly and

elevation with and without terrain correction applied. Also included are the C2 results from DTM and

EGM96 global geopotential model gravity anomalies.

The separation term C2 is positive in low land area, though very small (quasigeoid is relatively higher

than geoidal surface) and these surfaces reverse position with negative increase of C2 term as

elevation increases. It is negative in low land (Dadu area) in general, and become more negative in

high land (Kalat area). The correlation with elevation is more with observed data (94%) in Kalat area

than EGM96 model (49.8 %) values (Fig-4). However, this correlation is reversed in low land (Dadu

area) as C2 from observed data is less correlated (35.5 %) than EGM96 model (97%) separation term.

This also indicates an important aspect of inherent data deficiency in EGM96 model in these areas. In

low land area (Dadu) the EGM96 model with dominant low frequency components has close

correlation with low elevation and gravity gradient. In high land area (Kalat), the EGM96 model

values are ~50% less correlated than observed data indicating the deficient information regarding the

high frequency component of gravity and elevation.

R2C2EGM 96 = 0.498

R2C2Obs = 0.94

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 500 1000 1500 2000 2500 3000

Elevation (m)

Geo

id t

o Q

uasi

geoi

d C

oree

term

C2

(m)

Fig-4. Relationship of Elevation with geoid-quasigeoid correction C2 term from observedgravity data and EGM96 model gravity in the Kalat area

8

R2C2EGM 96= 0.97

R2C2Obs= 0.355

-3.00E-02-2.50E-02-2.00E-02-1.50E-02-1.00E-02-5.00E-030.00E+005.00E-031.00E-021.50E-022.00E-022.50E-02

0 50 100 150

Elevation(m)

Geo

id to

Qua

sige

oid

corre

ctio

Term

C2(

m)

The contour maps of the geoid to quasigeoid separation term from observed gravity data (Fig. 6 & 7

below) show an increase in absolute value in the north-west direction. The trend of the elevation is

also increasing towards north-west direction. Most of the western half and complete eastern half of

Dadu area is low elevation with 20-40 m above sea level, with the C2 term in the range of 2 -3 mm.

The Kalat area has highest elevation in central west part with corresponding maximum separation

term of 440 mm. This shows high correlation of C2 term with elevation. This also validates the data

quality of observed gravity and elevation in both study areas.

67.40 67.50 67.60 67.70

26.60

26.70

26.80

26.90

27.00

27.10

27.20

Longitude(degrees)

Latitud

e(de

gree

s)

Fig-5. Relationship of Elevation with geoid-quasigeoid correction C2 term fromobserved gravity data and EGM96 model gravity in the Dadu area

Fig-6. Geoid to Quasigeoid separation term C2 from observed gravity data in Dadu Area.Contour Interval=0.3mm

Latit

ude

(deg

rees

)

9

66.50 66.60 66.70 66.80 66.90 67.00

28.60

28.70

28.80

28.90

29.00

29.10

29.20

29.30

29.40

4-Conclusions and Recommendations:

The geoid to quasigeoid separation term C2 was estimated in both study areas and a comparative

study with elevation was established. There is a high correlation of the C2 term with elevation,

though the correlation of Bouguer anomalies and free air anomalies in the low land area is

comparable due to the very small Bouguer plate effect. The terrain correction effect in Kalat is

considerable and it is practically insignificant in Dadu area. Keeping in view of the general terrain of

Pakistan and the height datum of Pakistan, it is emphasized that the terrain correction be applied in

the calculation of geoid to quasigeoid separation term. Also, the magnitude of the geoid to quasigeoid

separation term C2 suggests its incorporation in the final geoidal solution, whichever course is

followed in solving the geodetic boundary value problem of the gravity field. This also necessitates

the estimation of C2 term over the whole area of Pakistan and its validation with other data sources

e.g. GPS/Leveling data.

Acknowledgements:Higher Education Commission of Pakistan is highly acknowledged for sponsoring this study under the

indigenous Ph.D. program at the Department of Earth Sciences Quaid-i-Azam University (QAU) Islamabad,

Pakistan. The Directorate General of Petroleum Concessions (DGPC), Ministry of Petroleum and Natural

Longitude (degrees)

Fig-7. Geoid to Quasigeoid separation term C2 from observed gravity data in Kalat Area. Contour Interval=10 mm

Latit

ude

(deg

rees

)

10

Resources, Islamabad, Pakistan is also acknowledged for providing the gravity data of two study areas to the

Department of Earth Sciences, QAU, Islamabad.

The authors are also very grateful to Professor Carl Christian Tscherning, University of Copenhagen, Denmark

for the constructive remarks on the initial manuscript of this paper.

Reference:Heiskanen, W. A. and Moritz, H. (1967) Physical Geodesy. Freeman, San Francisco.

Featherstone, W. E. and Dentith, M. C. (1994) Matters of gravity: the search for gold. GPS World5(7), 34-39.

Featherstone, W. E. and Dentith, M. C. (1997), A geodetic approach to gravity data reduction for

Geophysics. Computers & Geosciences Vol.23, No. 10, pp. 1063-1070.

Featherstone, W. E. and J.F. Kirby (1998), Estimates of the separation between the geoid and

quasigeoid over Australia. Geomatics Research Australia, No. 68, pp.79-90.

H. Nahavandchi (2002), Two different methods of geoidal height determinations using a spherical

harmonics representation of the geopotential, topographic corrections and height anomaly-

geoidal height difference. Journal of Geodesy, No. 76, pp. 345-352

Lemoine, F. G., Smith, D. E., Smith, R., Kunz, L., Pavlis, N. K., Klosko, S. M., Chinn, D. S.,

Torrence, M. H., Williamson, R. G., Cox, C. M., Rachlin, K. E., Wang, Y. M., Pavlis, E. C.,

Kenyon, S. C., Salman, R., Trimmer, R., Rapp, R. H. and Nerem, R. S. (1997). The

development of the NASA, GSFC and NIMA joint geopotential model, in Segawa,

Fugimoto and Okubo (eds), IAG Synzposza 117: Gravzly, Geozd, and Marzne Geodesy,

Springer-Verlag, Berlin, pp. 461-470.

Molodensky, M. S., Eremeev, V. F. and Yurkina M. I. (1962). Methods for the study of the external

gravitational field and figure of the Earth. Israeli Program for Scientific Translations,

Jerusalem.

National Imagery and Mapping Agency (1997). WGS 84 EGM96 (360,360) Public Web Page,

http://164.214.2.59/geospatial/products/GandG/wgs-84/egm96.html

Rapp. R. H. (1997). Use of potential coefficient models for geoid undulation determinations using a

spherical harmonic representation of the height anomaly/geoid undulation difference,

Journal of Geodesy 71: 282-289.

Sjoberg, L. E. (1995). On the quasigeoid to geoid separation, manuscripta geodaetica 20: 182.-192.

USGS, EROS, DATA Center, SIOUX Fall, SD 57198-0001,

http://srtm.usgs.gov/data/obtainingdata.htmal

Yecai Li and Michael G. Sideris (1993),Terrain Correction via 2 Dimensional Fast Fourier Transform

with either mass-Prism or mass-Line topographic model. Deptt. of Geomatics Engineering.

The University of Calgary, Canada.

Outlier detection in champ kinematic orbit data

to be used in gravity field determination

C. Gruber

Institute of Navigation and Satellite Geodesy, Graz University of Technology,

[email protected]

March 1, 2005

Abstract

champ orbit data have become available from dynamic, reduced dy-namic and kinematic (purely geometric) approaches. All of them havebeen used in gravity field determination. However, the kinematic orbitsenjoy higher acceptance since they do not employ any a-priori gravityfield model. Unfortunately, kinematic orbits suffer from higher noise lev-els and outliers. Hence, improving kinematic orbits will directly influencethe quality of the gravity field model. In this paper, a pre-processing strat-egy for kinematic orbit data will be presented. Based on orbit smoothing,outliers will be detected and excluded from further processing.

1 Introduction

The energy balance method is an established approach to determine the lowerspherical harmonic coefficients (in the case of champ up to degree/order ≈ 70)of the Earth gravity field (cf. Badura et al. 2004). Due to this approach thequality of the solution depends to a high degree on an accurate knowledge ofthe velocity of the spacecraft. If kinematic orbits shall be used, a strategy hasto be developed to derive those velocities with sufficient accuracy. In the caseof champ, where a geoid solution at dm precision is envisaged, the velocityrms must not be larger than 10−1

mm/s in order to meet this requirement. Forthe case of simulated orbits this can usually be achieved by standard numericaldifferentiation. Unfortunately, kinematic orbits suffer, compared to simulatedor reduced dynamic orbits from a much higher noise level due to low confidencein the gps configuration, i.e. poor satellite geometry or a low number of ob-servations for a certain epoch (cf. Svehla et al. 2003). Althought the overallorbital data is of unpreceedented quality, still outliers have to be recognized inorder to ensure the introduction of normally distributed kinetic energy as inputfor the balance equations.

1

The idea of how to detect outliers and to exclude them from further gravityfield processing is the correlation of the data by a simple model. Filtering theorbital data with the spectral properties of the model could then be used todefine a criterion for outliers.

2 The error model

We assume to estimate kinetic energy from orbit positions and their numericallyderived velocities as,

T =1

2

(

G2

R2+ r2

)

(1)

with G being the angular momentum, R the radius from the satellite to theEarth center and r the radial velocity. The expression in brakets in eq. (1) canbe assigned to q2, where q represents generalized momentum.

Understanding the force model as

U = −∫ r

r0

K(q)dq (2)

with q describing the shortest path from r0 to r in the Hamiltonian sense.

The total force K can be approximated as

K = ∇V⊕ + ∇V$ + F s (3)

with the conservative gradients of the Earth potential field, as well as of thirdbodies and the surface force F s, representing energy dissipation in the upperatmosphere. The surface force can be further analyzed by

F s = ac + b (4)

leading to a linear correction of the (in the case of champ) mismeasured dis-sipative accelerations ac. The unknown bias b can be approximated if thederivatives of the along track, cross track and radial velocities are set equal toeq. (3) cf. Gruber et al. (2005), or after integration along the orbit by applyinglow degree polynomials (Badura et al. 2004).

After determination of the dissipative forces a representive energy constant overa certain time span can be found by

E =1

n

n∑

i=1

(Ti + Ui) (5)

or by following ‖L‖1 and using instead of the mean value the median.

2

From interchanging eq. (5), a general reference momementum qr can be calcu-lated

qr =√

2(E − U(q)) (6)

Differences in position from the reference momentum and the change in positionper epoch ∆r of the satellite are obtained,

δr = qr∆t − ‖∆r‖ (7)

and can be treated by treshold values to define outliers.

3 Data preparation

The kinematic position data in the terrestial reference frame (itrf) has beentransformed into a non rotating frame and numerically differentiated. Fromanalysis of the inclination of the orbital plane, gross outliers can be easily de-tected with standard methods, e.g. regression based technics (cf. M-estimation,Huber 1973) and the covariance information for these positions has been ad-justed.

The gradient forces can be obtained from any a priori gravity field of theEarth as well as the solid Earth tides. The accelerometer data has been ro-tated into the orbital plane by the available attitude information as well as theorientation of the local orbital tripod in each point.

Fig. 1 shows the kinematic residuals from eq. (7) during a 12 hour time span(30 sec sampling), revealing a very good accordance between the kinematic orbitand the used model.

0 500 1000 1500

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

epochs

m

Figure 1: Kinematic position residuals between theused geopotential model and the measured data.

3

4 The Outlier detection

The kinematic residuals from eq. (7) will be split in the following by a suit-able filter into a smooth dynamic innovation, f(δr), with respect to the usedgeopotential coefficients of the reference field in eq. (2) and higher frequencycomponents g(δr), consisting of omitted geopotential spectrum, noise and out-liers. Instead of filtering in spectral domain by impulse response filters, wherehardly use of the available covariance information can be made, f(δr) shallbe approximated by a functional model with smooth characteristics in orderto approximate the continuous properties of the osculating ellipses. The termcontinuous in this context underlines the fact that small changes within thegeopotential field of the Earth will introduce only small changes to the orbitalgeometry of the spacecraft. Since the lower frequency part of the used referencefield, e.g. egm96 (Lemoine et al. 1998) is in good accordance to this by itsparameterization with spherical harmonics (sh), the occurence of spikes in thedata should be fully ascribed to outliers from the determination of the kinematicvelocities.

A similiar approach can be found in slr data processing, known as nor-mal point generator (Sinclair 1997). The range observations are reduced bypredicted satellite state vectors from force models and the differences are thenapproximated by smooth trend functions. The rms of the residuals to thetrendfunction will then be used as a rejection level for the observations.

4.1 Definition of a trend model

In our model the trend function is establised by harmonical base functions upto a finite spectral resolution and the corresponding amplitudes shall be deter-mined by an optimized fit to the data. In order to make the functional modelinsensitive to outlier peaks, the available covariance information will be used asan a priori observation dispersion as well as a robust estimation approach (cf.Koch/ Levenhagen 2002) that shall fit the trend function. Thus a spectrallylimited harmonic analysis of the kinematic residuals shall best approximate lowfrequencies as the innovative signal, and gain outliers as their signal counterpartsbeyond a to be defined treshold.

4.2 Estimation of the model parameters

The functional model can then be described by a Fourier series, but the coeffi-cients shall be determined by an estimation procedure in order to incorporatecovariance information.

f(δr) ⇔ F(ω) = a · cos(2πnτ) + ib · sin(2πnτ)

(a + i · b) = (A′Q−1A)−1A′Q−1f(δr) (8)

The linear parameter estimation, minimizing the norm of the residuals showsdeficiencies when outliers are in the data. We therefore applied an iterative

4

approach where assumed outliers are iteratively downweighted during the esti-mation process.

4.3 Definition of the spectral resolution of the trend model

In oder to supplement the low geopotential frequencies used in eq. (2) as wellas to account for unmodelled effects such as ocean tides, it has to be definedwhat spectral resolution of the trend function should be envisaged.

From analysis of Fig. 1 it can be seen that in order to remove a suitabletrend function a development up to a few principal frequencies would be suf-ficient. The corresponding frequency of the flattening term of the Earth (J2)transforms within a 12 hours time span, i.e. rougly 8 revolutions, to 16 har-monic cycles or an upper frequency of roughly (46 · 60)−1

Hz if the revolutionduration is ≈ 92min. If the trend function is limited to that degree, only apossible mismodeling of this particular geopotential frequency is being takencare of by the trend function.

If we regard the used gravity field model as a rough approximation only,then the trend function should be able to absorb deficiencies in all employedgeopotential frequencies in order to maintain independence from the model. Onthe other hand it is clear that during a given timespan only a limited numberof geopotential frequencies (e.g. mainly zonal terms) should truly affect thesatellites position. We therefore assume the isotropic commission error fromerror degree variances (rms) of the sh - spectrum as representative measure toanalyse the effect on the orbit positions. According to Kaula’s rule of thumb(Kaula 1966),

σl =

√

√

√

√

l∑

m=0

C2lm + S2

lm =√

2l + 110−5

l2(9)

approximates the spectral density (psd) of the fully normalized geopotentialcoefficients (C, S) and together with the transfer function

λl =GM

R

(a⊕

R

)l

(10)

we obtain error degree variances for the gravity potential at satellite height,

σUl=

√

λ2l σ

2l , (11)

with GM the gravity constant of the Earth, a⊕ mean Earth radius, l geopoten-tial degree, R geocentric radius of the satellite.

The equivalent error model according to Rapp (1978) reads,

σl(∆g) =

√

α1

l − 1

l + 1sl+2

1 + α2

l − 1

(l − 2)(l + 2)sl+2

2 (12)

5

where α1 = 3.404, α2 = 140.03, s1 = 0.998006, s2 = 0.914232

and the corresponding transfer function

λl =(ae

R

)l R

l − 1. (13)

Error propagation of σUlinto eq. (7) yields the reference position error per

degree

σl,δr =∆t

√

2(E − U(q))· σUl

(14)

due to the used geopotential model. Fig. 2 shows the position error per degree,comparing the two different error degree variance models, namely Kaula andRapp,

5 10 15 20 25 30 35 40−3

−2

−1

degree

m (l

og)

Kaula

Rapp 1979

Figure 2: position error per degree according to errordegree variances of the sh expansion from differentempirical models.

It is understood, that in order to approach a cm- level for the filtering of thekinematic position residuals the spectral range of the trend function should bein the case of Kaula’s rule equivalent to geopotential frequencies up to deg/ord20. On the other hand, the higher the resolution of the trend function becomes,the more difficult it gets to robustly estimate the corresponding parameters. Aproposed rigorous treatment would be the statistical assessment of the appliedmodel, i.e. testing for significance of the model parameters for each estimation.The outliers could then be statistically detected by data snooping. For details

6

cf. (Kern, 2005). In our approach we must admit a lot better quality to thereference field than the error psd of the coefficients indicate and therefore limitthe resolution of the trend function to an equivalent geopotential degree oflmax = 6. This can be understood in the following manner: the innovativedifference between the used reference model and the solution belonging to thekinematic orbit data shall be found far below the degree variance models ofKaula and Rapp. This is certainly a critical aspect and we might thereforeloose independend gravity field information within the spectral bandwith (bw)of the used reference geopotential model beyond (l > 6) that would possiblybe detected as outliers. Nevertheless, it turns out after geopotential recoveryof the filtered kinematic orbit (cf. section 5), the named spectral bw comesout different from the used reference model, inspite of being the weakest partin terms of infiltration of a- priori information. The system proofes to be stillcapable in recovering an independend solution. Fig. 3 shows the trend function,approximating the kinematic position residuals.

400 500 600 700 800 900−0.03

−0.02

−0.01

0

0.01

0.02

epochs

m

Figure 3: Fit of the trend function with a spatialresolution up to 15 epochs (deg/ord 6).

4.4 Definition of a treshold for outliers

Once the trend model has been fitted, the innovative part can be subtractedfrom δr, yielding the noise and outlier function g(δr). To distinguish outliersfrom remaining gravity field signal a treshold value has to be defined, takinginto account the remaining signal power that can be expected from the omissionerror of the geopotential development of the reference field,

∆δr =∆t

√

2(E − U(q))· ∆U (15)

7

where ∆U has been calculated by

∆U =

∫ r

r0

∇Vl,∞dq (16)

from egm96 but could be simultaneously obtained again from e.g. Kaula’s ruleof thumb. Fig. 4 shows the omission error according to eq. (15) from a signalbeyond deg/ord l > 40 in terms of orbit deviations.

0 500 1000 1500

−0.01

0

0.01

epochs

m

Figure 4: Range of the omission error in terms oforbit deviations

It is evident that the omission of geopotential signal beyond deg/ord 40represents a clear limitation for an outlier treshold. Reduction of the omissionerror by a newly trend reveals remaining deviations of less than 1 cm. Thetreshold value has therefore been set to 1.25 cm.

Fig. 5 shows the total situation after removal of the trend function. No as-sumeable geopotential information should be found beyond the treshold values.

8

600 650 700 750 800 850 900

−0.02

−0.01

0

0.01

0.02

0.03

epochs

m

dataomission error

Figure 5: Kinematic residuals after trend removaland omission error within a treshold.

5 Model test

After the filtering with egm96, modified error variances for the kinetic energyhave been introduced into a LS estimation of the full geopotential coefficientsfrom deg/ord 2 to 50, applying the energy balance equation (cf. Jekeli 1999).Fig. 6 shows the difference psd’s with respect to other gravity field models.

9

0 10 20 30 40 5010

−10

10−9

10−8

10−7

10−6

10−5

DEG/ORD

signal’s psdKaula’s rule∆ Grim5s1∆ EGM96∆ Eigen3s

Figure 6: Comparison of a filtered solution by egm96to existing geopotential models: grim5s1, egm96,eigen3.

It can be seen that the solution fits best to eigen3s, although not being usedfor the outlier detection. A similiar result is obtained if eigen3s shall be usedfor the filtering. If grim5s1 has been used, a slight deterioration of the resultcan be observed (not plotted) which is in accordance to the general deficiency ofthe approach in the bw above l > 6, as being stated earlier. In order to obtainacceptable results a good reference solution is thus necessary.

6 Conclusions and Outlook

The method presented in this paper is an alternative approach, compared to theuse of dynamic orbits, in order to detect outliers, by their kinematic correlationto external gravity field information.

In an ideal case, the a-priori covariance information would represent theconfiguration of the observation space. Projection onto the parameter spacewould then give an immediate result for the geopotential model. Unfortunatelythis is not the case and the available covariances give therefore a too optimisticpicture concerning the outliers.

Once a global gravity field solution has been processed, the a-posteriori co-variances for the velocities (and thus the positions) can be computed and theirdifferentiation being repeated. A question under investigation is, whether aniterative approach without use of any a-priori information will converge. Anevaluation strategy could moreover be defined, assessing the quality of the ap-proach and the error probability of successfully detection (or failure) of outliers.

10

7 Acknowledgement

The Author would like to thank the Geoforschungszentrum Potsdam (gfz) forproviding the accelerometry data as well as Drs. Drazen Svehla and MarkusRothacher who made available the kinematic orbit data. Valuable discussionsand input from Dr. Michael Kern has been gratefully appreciated.

The Authors Adress:Christian GruberGraz University of TechnologyInstitute for Navigation and Satellite GeodesySteyrergasse 30/III, A-8010 Graz, Austriaphone +43(0)316 873-6357fax +43(0)316 873-6845email [email protected]

11

References

Badura T., Gruber C., Klostius R., Sakulin C. champ gravity field processingapplying the Energy Integral Approach Joint CHAMP/GRACE Science Meet-ing,GFZ, July 6-8, 2004, IN: Arne K. Richter (ed.) Advances in Geosciences2004.

Gerlach C. et al., A champ-only gravity field model from kinematic orbits usingthe energy integral, Geophysical Research Letters, VOL. 30, NO. 20, 2037,doi:10.1029/2003GL018025, 2003.

Gruber C., Tsoulis D., Sneeuw N., champ Accelerometer calibration by meansof the equation of motion and an a-priori gravity field. Zeitschrift fur Vermes-sungswesen, ZfV, in print, 2005.

Huber P.J. Robust regression: Asymptotics, conjectures and Monte Carlo, Ann.Stat., 1, 799-821, 1973.

Jekeli C., The determination of gravitational potential differences from satellite-to-satellite tracking, Cel. Mech. Dyn. Astron., 75, 85-101, 1999.

Kaula W.M. Thoery of Satellite Geodesy, Applications of Satellites to Geodesy.Dover Publications, Inc. Mineola, New York (edition 2000).

Kern M., External Calibration and Validation Methods for the Satellite MissionGOCE, Final technical report in the frame of External ESA Fellowship at GrazUniversity of Technology 2005.

Rapp RH.: Potential Coefficient and Anomaly Degree Variance Modelling Re-visited. Tech. Rep. No. 293, The Ohio State University, Department of GeodeticScience, Columbus, Ohio 1979.

Reigber Ch., Luhr H., Schwintzer P.: First Champ Mission Results, SpringerVerlag, Berlin Heidelberg 2003

Sinclair A.T.: Data Screening and Normal Point Formation. Royal GreenwichObservatory, Cambridge, UK 1997.

Svehla D, Rothacher M, Kinematic and Reduced-Dynamic Precise Orbit Deter-mination of champ satellite over one year using zero-differences. EGS-AGU-EUG Joint Assembly 06-11 April 2003, Nice, France. Geophysical ResearchAbstracts, European Geophysical Society Vol. 5. ISSN:1029-7006

12

Latest Geoid Determinations for the Republic of Croatia

T. Bašić*, Ž. Hećimović**

* Faculty of Geodesy, University of Zagreb, HR-10000 Zagreb, Kačićeva 26, Croatia ** Av. M. Držića 76, 10000 Zagreb, Croatia

Abstract. Due to several improvements that have been done in comparison to the previous solution, like the usage of better terrestrial gravity data at the Adriatic Sea, much more GPS/leveling points available, wider topography integration area and better residual terrain modeling procedure, one-block collocation for entire area, and finally, four times denser computation grid, have resulted in the official geoid solution for Croatia HRG2000. This solution is based on the long wavelength gravity field structures from EGM96 global geopotential model.

Recent CHAMP and GRACE satellite missions are defining new standards in modeling of the Earth’s gravity field, improving global gravity field models especially in long- and medium-wave range. This was confirmed for the Croatian territory thanks to the undertaken comparison of several such models with GPS/leveling and HRG2000 geoid data.

Based on these facts and the availability of a new and much denser point gravity data set over the land area, newest geoid computation for the Republic of Croatia become possible. This paper offers a detailed description of the applied computation procedure, the geoid quality estimation using GPS/leveling points, and the presentation of specially developed computer program made for the purpose of geoid interpolation in any area of the state.

Keywords. Geoid, gravity, GPS/leveling, terrain modeling, satellite missions, HRG2000 geoid solution

1 Introduction

First serious attempts in determination of geoid surface in this region are connected with the establishment of independence of the Republic of Croatia. Nevertheless, a significant improvement was realized in 1998, when the first HRG98 solution was presented at the EGS 23rd General Assembly in Nice (Bašić et al. (1999)), and immediately after that HRG98A modified solution at the 2nd Joint Gravity and Geoid Meeting in Trieste (Bašić and Brkić (1999)).

During the year 2000 next step forward was made in the Department for Geomatics at the Faculty of Geodesy University of Zagreb, resulting with the most recent geoid solution HRG2000, see Bašić (2001). Since HRG2000 was proclaimed by the State Geodetic Administration as the official geoid surface of the Republic of Croatia, in the continuation the short presentation of this solution and the special computer program for interpolation purpose is given, followed by an overview of the latest efforts in the preparation for the calculation of the new solution HRG2005.

2 Computation of HRG2000 geoid 2.1 Previous investigations

In the frame of preparatory work (Hećimović (2001)), the numerical investigation with 14 global Earth Geopotential Models (EGM) was undertaken with the goal to find the model that best fits the Earth gravity field on the territory of Croatia. As reference values, GPS/leveling geoid undulations on 121 points were used. It was found out that global geopotential models EGM96 and GFZ97 fit best the Earth gravity field on our territory (see Table 1). In order to determine the existence of constant vertical shift between the Croatian vertical datum and EGM96 and GFZ97, two transformation models were defined. The transformation model which includes zero-undulation No fits better the real data and improves the existence of constant vertical displacement between GPS/leveling undulations and EGM96 and GFZ97 geoid of –1.37 m and -1.28 m respectively, for more details see Hećimović and Bašić (2003).

Another interesting investigation was the examination of the influence of the resolution change in reference Digital Terrain Model (DTM) on the calculation of short-wavelength effects (Residual Terrain Modeling - RTM) in gravity anomalies (Hećimović (2001), which showed that the use of 20′x30′ reference DTM yielded residual gravity anomalies with the best statistical properties to apply in the collocation (small and smooth residual gravity field).

Table 1. The statistics of differences between GPS/leveling and different EGM geoid undulations (in m).

Model Min Max Mean St. EGM96 -2.43 0.21 -1.33 0.44 GFZ97 -2.44 0.22 -1.21 0.47 OSU91A -4.43 2.07 -0.62 1.16 IFE88E2 -3.42 1.41 -0.55 0.93 GFZ93A -2.97 1.74 -0.62 0.85 GFZ93B -3.08 1.67 -0.69 0.85 GPM2 -4.45 2.08 -0.80 1.47 GRIM4 -3.37 2.28 -0.53 1.14 GEM-T3 -4.21 0.76 -1.65 1.11 JGM-1S -3.85 1.07 -1.99 1.00

2.2 Calculation method

The strategy for high-resolution local gravity field determination is using three parts of the gravity field information: the long-wavelength part is taken from the global geopotential model, the medium-wavelength part originates in terrestrial point gravity field observations like free-air gravity anomalies and GPS/leveling data, and the short-wavelength part is taken from the high-resolution digital terrain model. In the simple remove-restore technique, the reduced observations are thus written as linear functional of the anomalous gravity potential T (Bašić (1989)):

2

iRTMEGMi nTTTx +−−= )(L )(L )(L iii . (1)

The least squares collocation determines the approximation:

x)DC( C)(~ -1' += T

PPT . (2)

Here P is a point in space, matrix C contains the signal co-variances between the observations, CP contains the signal co-variances between the observations and predicted

'T~ value in the point P, and D is the variance-covariance noise matrix.

The least square collocation technique results in predictions )T~

(L j . To obtain the desired

results, the effect of the anomalous masses and the effect of the geopotential model need to be added back through the restore procedure:

)(L )(L )'~

(L )~

( jjj RTMEGMj TTTTL ++= . (3)

We decided to use collocation technique because the territory of Croatia is a relatively small area, so the huge and extensive numerical operations were possible to be done in one-step due also to their flexibility in handling heterogeneous irregular spaced data. In addition, we preferred to have the error estimates of predicted quantities (Bašić (1989)). 2.3 Data sets used

Since the global geopotential models EGM96 fit best the Earth gravity field on our territory (Table 1), it is used for definition of long-wavelength structures. This model consists of spherical harmonic coefficients complete to degree and order 360 (Lemoine et al. (1996)).

In performing the residual terrain modeling, three grids were put to use: the detailed model of topography 4"x5" (approx. 120x110 m), covering an area from 41° to 48° in latitude, and 12° to 21° in longitude, the coarse 5′x5′ grid of relief heights covering an bigger area from 32° to 57° and from 0° to 33° in latitude and longitude respectively, and 20′x30′ RTM reference grid of the same area as the coarse one. For the topographic masses the constant density of 2670 kgm-3 is assumed. These digital terrain models were applied in well-known Forsberg’s TC software (Forsberg (1984)) for computation of terrain effects on gravimetric quantities.

For the calculation of free-air gravity anomalies point gravity data has been applied over the continental part of Croatia, Slovenia and Bosnia and Herzegovina, as well as Hungary and Italy (but rare). The gravity anomalies over the Adriatic Sea have been derived in 5′x5′ grid from map 1:750 000 (Morelli et al. (1969)), while the data covering Serbia and Montenegro have been recalculated in 5′x5′ grid from Bouguer anomaly maps 1:200 000. In this way a gravity data bank with more than 7500 free-air anomalies was created (Bašić (2001)).

In Table 2 the main statistics of gravity anomalies is presented, where the effects of the applied remove procedure decreasing the standard deviation from 34.53 mgal for observed anomalies to 12.40 mgal for residuals (∆gOBS-∆gEGM96-∆gRTM) is evident. A significant reduction of the mean value from 8.70 mgal to 0.47 mgal (good centered data) can be recognized too (1 mgal = 10-5 ms-2).

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Table 2. Statistics of gravity anomalies and their residuals (in 10-5ms-2)

∆gOBS ∆gEGM ∆gRTM ∆gRES

Mean 8.70 14.27 -6.04 0.47St.Dev. 34.53 29.15 20.21 12.40Min -103.07 -95.59 -119.35 -47.04Max 180.14 109.33 106.80 64.31

A priori information about the variation of the local gravity field is introduced through

the empirical covariance function calculated using residual gravity anomalies. In our case the variance of the empirical covariance function has the value of 154.09 mgal2 and the first zero-value occurs already at the distance of 50 km (Bašić (2001)).

For the purpose of correct absolute orientation of the calculated geoid surface, a limited number (138) of GPS/leveling points distributed across the Croatia has been used. The statistics are presented in Table 3, where an apparent remove effect is present again, but it should be noted that the value of the mean NRES = -2.16 m, most likely originates from the discrepancy between the used EGM96 model and the definition of national vertical datum, which is related to the tide gauge in Trieste. Table 3. The statistical characteristics of geoid reduction effect in 138 GPS/leveling points (in m)

NGPS/l NEGM9 NRTM NRES

Mean 44.41 45.78 0.81 -2.16St.Dev. 1.22 1.07 0.23 0.33Min 39.65 40.35 0.61 -3.18Max 46.88 47.63 1.67 -1.50

2.4 Geoid prediction

The actual computation area has been chosen to cover the entire territory of Croatia: from 42.0° to 46.6° in latitude, and 13.0° to 19.5° in longitude, with calculation grid of 1′x1.5′ (approx. 1.8x2.0 km) or total 72 297 prediction points. As a final product selected HRG2000 with 36 184 points was defined covering strictly the Croatian territory (Fig. 1). Table 4 gives the main statistics for the selected HRG2000 geoid, their error estimates and belonging height information. For the predominant part of selected area the geoid undulations have the standard deviations of 1-2 cm and only in the Central Adriatic Sea they are up to 10 cm (Bašić (2001)).

The comparison of HRG2000 with former HRG98 solution (Bašić et al. (1999) shows significant differences resulting from the application of more reliable gravity anomalies over the Adriatic Sea, considerably more GPS/leveling points (138), better residual terrain modeling (20′x30′ reference DTM and farther integration up to 2000 km), as well as successful realization of one step collocation.

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Furthermore, calculation grid in HRG2000 solution is four times denser. Although we met dramatically huger demands in numerical processing of topography data as well as in collocation, everything was successfully done on an average personal computer.

Fig. 1 Selected HRG2000 geoid surface (m). Table 4. Statistics for selected HRG2000 geoid undulations, their standard deviations and corresponding heights (36184 data, in m)

N St. Dev. HHRG2000

Mean 43.2 0.02 118.9St.Dev. 1.9 0.02 347.9Min 36.3 0.01 -1169.0Max 47.0 0.11 1781.0

3 Computer program for interpolation

Resulting from an increasing number of GPS-technology users in Croatia, a special scientific and professional project has been made by the State Geodetic Administration and

5

the Department for Geomatics at the Faculty of Geodesy that resulted with the computer program: IHRG2000. The program is written in Microsoft Visual Basic 6.0 and supports all the latest Windows platforms. The basic purpose of the program is the interpolation of HRG2000 geoid (Fig. 1) and the presentation of the results on screen, their storage on a disc and a print out. In the IHRG2000 program there are two possibilities of interpolation: bilinear and spline (Bašić and Šljivarić (2003)).

After initiating the program IHRG2000, the initial form in Croatian language is shown on the screen (Fig. 2). The program is ready for data input and for the changes in computation parameters. The most important input values are latitude and longitude in DEG (degrees and decimal degree parts) or DMS (degree, minutes and seconds) units. They are entered in three ways: by keyboard, disc, and from a map.

Fig. 2 Initial form of IHRG2000 program

The utility programs IHRG2000, created in the Department for Geomatics, to be used by the State Geodetic Administration of the Republic of Croatia should find their application in the practice, especially because the application of modern geodetic technologies requires that. Without knowing the accurate geoid surface it is impossible to make the connection between ellipsoid heights that are today very accurately provided by GPS technology, and orthometric heights that are used in practice. The licensing system of the program by the State Geodetic Administration provides the uniqueness and official character for this computer program, as well as for the data processed and realized using this program. 4 Preparations for the calculation of the new geoid

In 2004 the work was continued on finding even better solution for the geoid surface in Croatia. Within the frame of these efforts, the analysis of CHAMP and GRACE geoid solutions on our territory was made (Hećimović and Bašić (2004a)), the new digital terrain model Shuttle Radar Topography Mission (SRTM) was used for computing topographic effects of the Earth gravity field and compared with the so far used DTM (details in Hećimović and Bašić (2004b)), and the most important of all, the situation in gathering and quality control of the new gravity data has been significantly improved in this area.

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4.1 CHAMP and GRACE geoid models

Recent CHAMP and GRACE satellite missions define new standards in modeling gravity field of the Earth. Gravity signals that were out of sensitivity band of previous measurement techniques are becoming clearly recognized. CHAMP and GRACE are improving global gravity field models especially in long-wave and middle-wave range.

To estimate how well CHAMP and GRACE geopotential models fit gravity field in Croatia, the comparison of seven CHAMP and three GRACE models with GPS/leveling undulations (121 points) and HRG2000 geoid in 1'x1' raster has been made. GRACE models show better fitting of gravity field than CHAMP models (see Tables 5 and 6). The differences from gravity field with CHAMP models are strongly correlated with topography. The differences of GRACE gravity models are containing higher amount of short-wave topography gravity signal than CHAMP models, but the discrepancies from a more precise topography structure can be recognized. GRACE model GGM01C is the best fitting gravity field (Fig. 3 and Table 6), details in Hećimović and Bašić (2004a). Table 5. The main statistical characteristics of differences between GPS/leveling and recent global geoid models (121 points; in m)

Model Min Max Mean St. Dev. EIGEN-2 -4.70 2.30 -1.34 1.35 EIGEN-3p -3.40 1.89 -1.07 1.04 TUM-1S -3.74 1.57 -1.26 1.15 TUM-2Sp -4.16 1.98 -0.62 1.29 ITG-CHAMP01E -3.50 1.88 -0.91 1.07 ITG-CHAMP01S -4.11 1.72 -0.89 1.13 ITG-CHAMP01K -4.13 1.69 -0.88 1.16 GRACE01S -2.51 1.13 -1.08 0.80 GGM01S -2.59 1.33 -1.06 0.82 GGM01C -2.02 0.17 -1.00 0.45

Obtained wavelengths of differences are in short wave ranges depending on the terrain

behavior. In the higher mountain area the differences are bigger than in the flat area. As the new CHAMP and GRACE global geopotential models represent much better the long and medium wave gravity field structures at the Croatian territory than the older models, they could certainly be used for the computation of the new geoid solutions in the future. Table 6. The main statistical characteristics of differences between global geoid models and HRG2000 (53135 grid points; in m)

Min Max Mean St. Dev. EIGEN2 - HRG2000 -0.51 5.74 2.54 1.61 GRACE01S - HRG2000 -0.99 3.38 1.23 0.78 GGM01S - HRG2000 -1.20 3.72 1.18 0.83 GGM01C - HRG2000 -0.29 2.50 1.20 0.46

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Fig. 3 Differences between GGM01C and HRG2000 geoid solutions (m)

4.2 New Gravity Data Base

In the meantime, a new gravity data base with over 46700 items has been successfully established, out of which over 41000 point gravity values cover the land part of former Yugoslavia (see Fig. 4).

We deal here with essentially larger number of available gravity data than it has been the case so far (more than 10 times). Using the method of prediction by least squares, the quality of these data was first checked comparing all measured gravity values with those predicted on the basis of adjacent points. In Fig. 5 one can see the points where the differences were obtained between the measured and predicted anomalies being larger than the three times of standard deviations (206 differences up), i.e. larger than the two times of standard deviations (1372 differences down), both obtained from the prediction. It is obvious that a very small number of measurements are problematic in Croatia, and the majority of such measurements are outside of our area where an additional analysis of these data will be necessary in order to find the causes.

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Fig. 4 New available gravity data base

Fig. 5 Differences between measured and predicted gravity anomalies (left 206 diff. > 3·std, and right 1373 diff. > 2·std, obtained from the preduction)

5 Conclusion

Since we are now in the situation that most preparatory work has been done, determination of the new geoid solution for the territory of Croatia is expected in the nearest future, using also other methods of calculation, like FFT and integral formulas. We are also happy to be involved in the international European Gravity and Geoid Project (within IAG Commission 2) serving as a regional data and computing center. Therefore, we would like to develop a good cooperation with all surrounding countries.

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References

Bašić, T. (1989). Untersuchungen zur regionalen Geoidbestimmung mit “dm” Genauigkeit, disertation (in German), Wiss. Arbeit. der Fachrichtung Vermess. der Univ. Hannover, Nr. 157.

Bašić, T. (2001). Detailed Geoid Model of the Republic of Croatia (in Croatian), Reports of the State Geodetic Administration on Scientific and Professional Projects in the Year 2000, Ed. Landek I., 11-22, Zagreb.

Bašić, T., M. Brkić, and H. Sünkel (1999). A New, More Accurate Geoid for Croatia. EGS XXIII General Assembly, Nice, 20-24 April 1998, In: Physics and Chemistry of the Earth, Part A: Solid Earth and Geodesy, Special Issue: Recent Advances in Precise Geoid Determination Methodology, I. N. Tziavos and M. Vermeer (eds.), pp. 67-72, Elsevier Science Ltd.

Bašić, T., and M. Brkić (1999). The Latest Efforts in the Geoid Determination in Croatia, 2nd Joint Meeting of the International Gravity Commission and the International Geoid Commission, 7-12 September, 1998, Trieste, Bollettino di Geofisica Teorica ed Applicata, Vol. 40, N. 3-4, pp. 379-386, Osservatorio Geofisico Sperimentale.

Bašić, T., and M. Šljivarić (2003). Utility Programs for Using the Data of the Official Croatian Geoid and Coordinate Transformation between HDKS and ETRS (in Croatian), Reports of the State Geodetic Administration on Scientific and Professional Projects from the Year 2001, Ed. I. Landek), 21-32, Zagreb.

Forsberg, R. (1984). A Study of Terrain Reductions, Density Anomalies and Geophysical Inversion Methods in Gravity Field Modeling, Dep. of Geod. Sci. and Surv., Rep. No. 355, Ohio State University, Columbus.

Hećimović, Ž. (2001). Modeling of Reference Surface of Height Systems, Ph.D. thesis (in Croatian), 1-135, Faculty of Geodesy, University of Zagreb.

Hećimović, Ž., and T. Bašić (2003). Global Geopotential Models on the Territory of Croatia (in Croatian). Geodetski list, 57 (80), No. 2, 73-89, Zagreb.

Hećimović, Ž., and T. Bašić (2004a). Comparison of CHAMP and GRACE Geoid Models with Croatian HRG2000 Geoid, 1st General Assembly of the European Geosciences Union, Session G9: Results of Recently Launched Missions: Champ, Envisat, Icesat, Jason-1, and Grace, Nice, France, 25-30 April 2004, poster EGU04-A-02715; G9-1WE3P-0316, Abstract in: Geophysical Research Abstracts (CD), Volume 6, 2004, ISSN 1029-7006.

Hećimović, Ž., and T. Bašić (2004b). Modeling of Terrain Effect on Gravity Field Parameters in Croatia, IAG International Symposium Gravity, Geoid and Space Missions (GGSM2004), 30. Aug. – 3. Sep. 2004, Porto, Portugal.

Lemoine F. G., et al. (1996). The Development of the NASA, GSFC and NIMA Joint Geopotential Model, Proceed. paper for the IAG Symposium on Gravity, Geoid, and Marine Geodesy, Tokyo.

Morelli, C., M.T. Carrozzo, P. Ceccherini, I. Finetti, C. Gantar, M. Pisani, P. Schmidt di Friedberg (1969). Regional Geophysical Study of the Adriatic Sea, Bollettino di Geofisica Teorica ed Applicata, Vol. XI, N. 41-42, Trieste.

10

Terrain Effect on Gravity Field Parameters using Different Terrain Models

Ž. Hećimović*, T. Bašić**

* Av. M. Držića 76, 10000 Zagreb, Croatia ** Department for Geomatics, Faculty of Geodesy, University of Zagreb, HR-10000 Zagreb, Kačićeva 26, Croatia

Abstract. This paper presents the results of several analyses considering influence of terrain models on gravity field data. In the analysis are used digital elevation model (DEM) made by digitalization of topographic maps, Shuttle Radar Topography Mission Digital Terrain Elevation Data (SRTM DTED), gravity anomalies, GPS\leveling undulations and deflections of the vertical in Croatian geoid test area. To check the influence of different terrain resolutions on residual terrain model (RTM) effects on gravity anomalies and GPS\leveling undulations, the resolutions of referent terrain models 6’x7.6’, 10’x15’, 20’x30’ and 30’x30’ are used. Influence of different referent DEM resolutions on remove-restore residual fields is showing the smoothest and smallest field for resolution 20’x30’. The comparison of DEM and SRTM DTED terrain models is made to check their quality. To judge influence on gravity field considering differences of DEM and SRTM DTED terrain models, RTM effects on gravity anomalies, GPS\leveling undulations and vertical deflections are modeled.

Keywords. DEM, SRTM DTED, terrain model resolution, RTM effect.

1. Introduction

CHAMP and GRACE and further GOCE gravity models are defining new standards in gravity field modeling in global (long wave) and regional (middle wave) scale. Topographic, short wave, influence on gravity is becoming more and more important. However, the latest GRACE models are containing topographic (short wave) structure of gravity field, see Hećimović and Bašić (2004). Different characteristics of Croatian topography, on the coast very slope, high mountains and in the eastern part flat area, can be clearly recognized in the GRACE models, see Hećimović et al. (2004). Modeling of short wave gravity field, which is caused by topography and masses densities variability, is becoming the primary problem in modeling regional and local gravity field. To investigate influence of different terrain models on gravity anomalies, deflections of the verticals and GPS\leveling undulations the test area in the northern part of Croatia has been chosen. To get a insight into the terrain resolution influences, it is interesting to analyze the influences of different terrain model resolutions on RTM gravity anomalies and RTM GPS\leveling undulations. Terrain models with the resolutions 6’x7.6’, 10’x15’, 20’x30’ and 30’x30’ are used in investigations. For every resolution, the statistical characteristics are analyzed as well the main characteristics of empirical covariance functions. Besides the used DEM that is based on national topographic information, global topographic information, e.g. SRTM DTED is also publicly available now. Comparing of DEM and SRTM DTED we obtain the independent quality control of available topographic information. Calculating RTM influences on gravity functionals using DEM and SRTM DTED gives us the possibility to check direct influence on gravity field values as well on geoid using independent sources of topographic information. It gives us the possibility to judge systematic errors in topographic models on gravity field values.

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2. Croatian Geoid Test Area

Croatian political borders have got an unpleasant shape for modeling earth gravity field. The rectangular test area (45.0° < ϕ < 46.5°, 15.5° < λ < 17.5°) is chosen for numerical analysis (see Figure 1).

Fig. 1 Croatian geoid test area with point free air anomalies (green points), deflections of the verticals (blue crosses) and GPS\leveling data (red triangles). There are point free air anomalies, deflections of the verticals, GPS\leveling data and different terrain models used in the analysis. Because of inconvenient Croatian shape for modeling earth gravity field, the data about the territory of other countries are also used. To separate different gravity fields’ wavelengths, and analyze smaller gravity signal, the remove-restore procedure on gravity field functionals Li is used (Denker (1988)). Wavelengths are separated considering different sources of data after

. (1) )RTMT(iL)MT(iL)GPMT(iL)T(iL ++=

The theoretical background to mode RTM effect can be found in Forsberg (1984a) or Forsberg and Tscherning (1981). The main statistical characteristics of 1491 point free air anomalies are given in Table 1. Table 1. The main statistical characteristic of gravity anomalies

∆gFREE AIR

[mGal1] ∆g EGM96

[mGal] ∆g RTM

[mGal] ∆g REZID.

[mGal] Mean 31.85 23.76 -0.86 12.70 St. dev. 23.46 9.11 10.31 12.76 Min. -40.76 -3.06 -43.88 -22.24 Max. 135.22 47.51 74.61 64.26

From available GPS\leveling dataset, only 41 reliable GPS\leveling undulations in the test area are used. The main statistical characteristics of GPS\leveling data are shown in Table 2. 1 1 mGal = 1·10-5 m·s-2

2

Table 2. The main statistical characteristic of GPS\leveling undulations

NGPS\LEV.

[m] N EGM96

[m] N RTM

[m] N REZID.

[m] Mean 45.16 46.46 -0.01 -1.29 St. dev. 0.48 0.38 0.05 0.26 Min. 44.03 45.71 -0.07 -1.83 Max. 46.44 47.07 0.14 -0.68

RTM undulations are indicating small influence on the geoid height changes, and the mean of undulations is indicating datum differences in the first approximation. Deflections of the vertical components are used in 91 points. They are behaving irregularly in amplitude and directions. The main statistical characteristics of deflections of the vertical components are given in Tables 3a and 3b. Table 3a. The main statistical characteristic of meridian component of deflections of the vertical

ξ MEAS. [”]

ξ EGM96 [”]

ξ RTM [”]

ξ REZID. [”]

Mean 0.64 0.29 0.02 0.33 St. dev. 3.61 1.64 1.28 2.35 Min. -10.46 -4.96 -2.92 -7.16 Max. 11.02 2.95 4.70 6.18

Table 3b. The main statistical characteristic of longitude component of deflections of the vertical

η MEAS. [”]

η EGM96 [”]

η RTM [”]

η REZID. [”]

Mean 1.55 1.32 -0.29 0.52 St. dev. 3.07 1.61 1.57 2.11 Min. -10.18 -4.85 -5.36 -4.62 Max. 7.56 4.75 5.76 4.96

The mean values for gravity anomalies and undulations are indicting datum differences that should be considered before geoid modeling. 3. Terrain Models

The terrain models are used for modeling of terrain influence on gravity field DEM of different resolutions and new SRTM DTED. Fine DEM is made by digitizing the maps in the scale 1:25000. Rough and referent terrain models are made by resampling of the fine DEM model. SRTM DTED for area 12 (USGS EROS Data Center (2004)) in the resolution 3”x 3” is used as a separate fine terrain model. Coverage of terrain models is shown on Figure 2.

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Fig. 2 Terrain models. - fine SRTM DTED, 3”x 3” in area 44.5° < ϕ < 47.0°, 14.0° < λ < 19.0°, - fine DEM, 4”x 5” in area 44.5° < ϕ < 47.0°, 14.0° < λ < 19.0°, - rough DEM, 2.0’x 2.5’ in area 40.0° < ϕ < 49.0°, 10.0° < λ < 24.0°, - referent DEM, 10’x 15’ in area 40.0° < ϕ < 49.0°, 10.0° < λ < 24.0°. 4. Analysis of Influence of Terrain Model Resolution Changes on Gravity Field Functionals

To judge the influence of terrain model resolution changes on modeling RTM effect, a resampling of referent DEM is made. In the analysis there are resolutions 6’x 7.5’, 10’x 15’, 20’x 30’ and 30’x 30’ used. RTM effect on gravity anomalies and GPS\leveling undulations is modeled using TC program (Forsberg (1984b)). Constant terrain density of 2670 kg·m-3 is used. The main statistical characteristics for RTM gravity anomalies and RTM GPS\leveling undulations are given in Tables 4 and 5. Table 4. The main statistical characteristics of RTM gravity anomalies using different referent DEM resolutions

6’ x 7.5’ ∆g RTM

[mGal]

10’x15’ ∆g RTM

[mGal]

20’x30’ ∆g RTM

[mGal]

30’x30’ ∆g RTM

[mGal] Mean -3.72 -4.17 -4.62 -8.59 St. dev. 11.42 13.47 15.67 19.12 Min. -77.06 -74.58 -65.98 -74.40 Max. 61.31 77.69 89.98 92.57

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Table 5. The main statistical characteristics of RTM GPS\leveling undulations using different referent DEM resolutions

6’ x 7.5’ N RTM

[m]

10’x 15’ N RTM

[m]

20’x 30’ N RTM

[m]

30’x 30’ N RTM

[m] Mean 0.01 -0.01 -0.03 -0.50 St. dev. 0.03 0.07 0.15 0.67 Min. -0.08 -0.23 -0.40 -1.91 Max. 0.31 0.57 0.89 0.95

Increasing resolution of the referent DEM is causing the increase of standard deviations of RTM gravity anomalies and RTM GPS\leveling undulations. With increasing referent DEM resolution the mean values of RTM gravity anomalies are increasing regularly, and the data are deflecting from centering conditions. The mean values of RTM GPS\leveling undulation are increasing regularly up to the resolution 30’x30’, where big jump can be observed. 5. Analysis of Influence of Terrain Model Resolution Changes on Empirical Covariance Function

Covariance function is essential for gravity field modeling in collocation techniques. To judge the influence that the changing of referent DEM resolution makes on empirical covariance function of free air anomalies, residuals of free air anomalies are calculated with different RTM values considering changing of resolutions of DEM. In Table 6 the main statistical characteristics of gravity anomalies residual fields are presented. Table 6. The main statistical characteristics of free air anomalies residual fields in the middle wavelength calculated using different resolutions of referent DEM

6’ x 7.5’ ∆gM

[mGal]

10’x15’ ∆gM

[mGal]

20’x30’ ∆gM

[mGal]

30’x30’ ∆gM

[mGal] Mean 11.85 12.22 12.67 16.64 St. dev. 18.37 15.68 12.76 15.66 Min. -48.15 -27.38 -22.27 -21.15 Max. 90.77 72.92 64.19 81.08

The residual anomalies calculated by RTM effect with 20’x30’ referent DEM resolution are giving the smoothest and the smallest residual fields. The mean values of residuals are bigger because of the consistency of gravimetric datums. The associated variance for the same referent DEM resolution is also the smallest one (see Table 7).

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Table 7. Variances of residual free air anomalies for different resolutions of referent DEM

DEM resolution

Variance [mGal2]

6’ x 7.5’ 407.725 10’ x 15’ 368.381 20’ x 30’ 308.050 30’ x 30’ 535.461

Using these statistical data as criteria, referent DEM of resolution 20’x30’ gives the best results, e.g. this referent DEM gives in remove-restore procedure the smallest and the smoothest residual field for geoid modeling. 6. SRTM Digital Terrain Elevation Data

NASAs Shuttle Radar Topography Mission (SRTM) made in 11 days Space Shuttle flight has scanned nearly a global Earths topography. Space borne Imaging Radar-C (SIR-C) sensors formed an interferometer with a 60-meter long baseline. The data have the potential to provide almost global elevation data in up to 30 m resolutions. SRTM Digital Terrain Elevation Data (DTED) are unrestricted for the area anywhere on the globe in 3” (~100m) resolution, but the resolution 1” (~30m) is publicly available only for the United States and its territories. SRTM DTED terrain data have voids mainly caused by SIR-C radar shadows in rough terrain. To fill SRTM DTED voids GTOPO30 the terrain model is used (see Figure 3).

Fig. 3. a) Source SRTM DTED with voids (white holes) and b) SRTM DTED terrain model treated with GTOPO30. The problem in using SRTM DTED terrain model is that it is related to radar reflecting surface. It does not have to be the primary terrain, but the top of the trees and buildings respectively. 7. Comparison of DEM and SRTM DTED Terrain Models

To judge the quality of DEM and SRTM DTED terrain models in test areas, SRTM DTED is resampled in fine DEM raster of 4”x5”. The main statistical characteristics of DEM and SRTM DTED terrain models and their differences are shown in Table 8, and differences are presented on Figure 4.

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Table 8. The main statistical characteristics of DEM and SRTM DTED terrain models and their differences

H DEM

[m]

H SRTM DTED

[m]

Diff. H DEM -

H SRTM DTED

[m] Mean 229 228 1 St. dev. 181 182 27 Min. 1 44 -309 Max. 1796 1770 357

Fig. 4 DEM and SRTM DTED terrain model differences [m]. DEM and SRTM DTED differences have mean value of 1 m. It indicates, in the first approximation, terrain model datum differences. It is caused by datum differences between old Croatian height system datum defined by costal tidal station in Trieste and global geoid EGM96. This value is showing good agreement with previous comparisons of Croatian height system and global geopotential models, see Hećimović (2001), Hećimović and Bašić (2002). In these investigations there are 121 GPS\leveling undulations well distributed over Croatian territory used. They showed differences between Croatian height system datum and global geoid EGM96 of 1,37 m, and that value has good agreement with the value obtained with DEM and SRTM DTED in table 8. 8. Influence of DEM and SRTM DTED Differences on RTM Gravity Functionals

To check influence of different topography data solutions on gravity field, RTM effects are modeled using DEM and SRTM DTED terrain models. The main statistical characteristics of RTM gravity anomalies for DEM and SRTM DTED terrain models and their differences are presented in Table 9, and on the Figure 5 the differences can be seen.

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Table 9. The main statistical characteristics of RTM gravity anomalies for DEM and SRTM DTED terrain models and their differences

∆g RTM DEM

[mGal]

∆g RTM

SRTM DTED

[mGal]

Diff. ∆g RTM DEM –

∆g RTM SRTM DTED [mGal]

Mean -4.74 -4.29 0.13 St. dev. 12.44 13.52 0.83 Min. -74.58 -74.58 -7.74 Max. 67.37 78.91 6.36

Differences have small bias of 0.13 mGal, as well small dispersion, e.g. standard deviation.

Fig. 5 Differences of DEM and SRTM DTED RTM gravity anomalies [mGal]. The main statistical characteristics of RTM GPS\leveling undulations from DEM and SRTM DTED terrain models and their differences are given in Table 10. The main statistical characteristics and Figure 6 indicate that the influence of different terrain models on RTM GPS\leveling undulations is small. Table 10. The main statistical characteristics of RTM GPS\leveling undulations for DEM and SRTM DTED terrain models and their differences

NRTM DEM

[m]

NRTM

SRTM DTED [m]

Diff. NRTM DEM -

NRTM SRTM DTED [m]

Mean -0.01 -0.01 -0.00 St. dev. 0.06 0.06 0.01 Min. -0.07 -0.08 -0.02 Max. 0.14 0.15 0.01

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Fig. 6 Differences of DEM and SRTM DTED RTM GPS\leveling undulations [m]. RTM meridian component of deflections of the vertical of DEM and SRTM DTED terrain models and their differences (see Table 11 and Figure 7) are showing small influence of different terrain models. On the Figure 7 it can be seen that the influence is increasing in higher areas. Table 11. The main statistical characteristic of RTM meridian component of deflections of the vertical for DEM and SRTM DTED terrain models and their differences

ξ RTM DEM

[”]

ξ RTM SRTM DTED

[”]

ξ RTM DEM - ξ RTM SRTM DTED

[”] Mean 0.02 0.32 -0.30 St. dev. 1.28 1.40 0.56 Min. -2.92 -2.33 -2.83 Max. 4.70 4.99 0.45

Fig. 7 Differences of DEM and SRTM DTED RTM meridian component of deflections of the vertical [arc sec].

Similar as by RTM meridian component, RTM longitudinal component of deflections of the vertical of DEM and SRTM DTED terrain models and their differences (see Table 12 and Figure 8) show small influence of different terrain models. A small increase of influence can be recognized in higher areas.

9

Table 12. The main statistical characteristic of RTM longitude component of deflections of the vertical for DEM and SRTM DTED terrain models and their differences

η RTM DEM

[”]

η RTM SRTM DTED

[”]

η RTM DEM - η RTM SRTM DTED

[”] Mean -0.29 -0.04 -0.25 St. dev. 1.58 1.48 0.56 Min. -5.35 -5.03 -2.52 Max. 5.76 6.17 0.58

Fig. 8 Differences of DEM and SRTM DTED RTM longitudinal component of vertical deflections [arc sec]. 9. Conclusions

The influence of different referent terrain model resolutions on RTM effect for gravity anomalies and GPS\leveling undulations is analyzed for resolutions 6’x7.6’, 10’x15’, 20’x30’ and 30’x30’. The increasing of DEM resolutions influences RTM gravity anomalies standard deviations from 11.42 mGal to 19.12 mGal and mean values from -3.72 mGal to -8.59 mGal. The mean values of RTM GPS\leveling undulations change along with the increasing terrain resolution from 0.01 m to -0.50 m, and standard deviations from 0.03 m to 0.67 m. The changes for RTM undulations are regular up to resolution 30’x30’ where a big jump can be observed. Despite of this irregularity, influence of different resolutions on RTM effect is significant and regular. The changing of referent DEM resolution affects the covariance function and modeling of gravity signal in collocation technique. Anomalies have the smallest variance for 20’x30’ terrain resolution and residual field of gravity anomalies has the best results in comparison with the results when other terrain resolutions are used for modeling RTM effects. Terrain model of this resolution is giving the smoothest and the smallest residual field for modeling gravity field. Considering the criteria of the smallest variance and the best main statistical characteristic of remove-restore residual field, the optimal resolution of DEM can be found. The smoothest and the smallest residual gravity field are wanted by gravity field modeling using collocation technique. To judge quality of different terrain models, DEM and SRTM DTED models are compared in the test area. They have vertical bias of 1 m, that is indicating datum differences in the first approximation, and the standard deviation, which is characterizing dispersion of terrain models differences, is 27 m.

10

11

To analyze the influence of differences of DEM and SRTM DTED terrain models on gravity field functionals, RTM-effects on gravity anomalies, GPS\leveling undulations and vertical deflections are modeled, and differences are analyzed. Differences of DEM and SRTM DTED terrain models are causing RTM effect on gravity anomalies that have the mean value of 0.13 mGal and standard deviation of 0.83 mGal, RTM GPS\leveling undulations have the mean value of 0.00 m and standard deviation of 0.01 m, meridian vertical deflection components have the mean value of -0.30” and standard deviation of 0.56” and longitudinal vertical deflection component have the mean value of -0.25” and standard deviation of 0.56”. DEM and SRTM DTED terrain models discrepancies cause small differences in geoid modeling data, but they should be considered. The problem that SRTM DTED is related to radar reflecting surface and not to terrain, should be considered in geoid modeling with centimeter accuracy. References

Denker, H. (1988). Hochauflösende regionale Schwerefeldbestimmung mit gravimetrischen und topographischen Daten. Wissenschaftlichen Arbeiten der Fachrichtung Vermessungswesen der Universität Hannover, Nr. 156, Hannover

Forsberg, R., C. C. Tscherning (1981): The Use of Heights Data in Gravity Field Approximation. Journal of Geophysical Research (86), 7843-7854

Forsberg, R. (1984a). Local Covariance Functions and Density Distributions. The Ohio State University, Department of Geodetic Science and Surveying, Report No. 356, Ohio

Forsberg, R. (1984b). A Study of Terrain Reductions, Density Anomalies and Geophysical Inversion Methods in Gravity Field Modeling. Report of the Department of Geodetic Science and Surveying, Report No. 355, Ohio

Hećimović, Ž. (2001): Modeliranje referentne plohe visinskih sustava. Dissertation. University of Zagreb Faculty of Geodesy (in Croatian)

Hećimović, Ž., T. Bašić (2002). Global geopotential models on the territory of Croatia. Geodetski list, 57 (80), Nr. 2, 73-89 (in Croatian)

Hećimović, Ž., T. Bašić (2004). Comparison of CHAMP and GRACE geoid models with Croatian GRG2000 geoid. 1st General Assembly European Geosciences Union (EGU), Nice, France, from 25 - 30 April 2004. Poster presentation. Geophysical Research Abstracts, Vol. 6, 02715, 2004. SREef-D: 1607-7962\EGU04-A-02715. European Geosciences Union

Hećimović, Ž., B. Barišić, I. Grgić (2004). European Vertical Reference Network (EUVN) considering CHAMP and GRACE gravity models. Symposium of the IAG Subcommission for Europe. European Reference Frame - EUREF 2004, Bratislava, Slovakia, 2 - 5 June 2004

USGS EROS Data Center (2004). SRTM DTED Level 1, Area 12, CD-ROM

1

Identifying sea-level rates by a wavelet-based multiresolution analysis of

altimetry and tide gauge data

E.V. Rangelova1, R.S. Grebenitcharsky2 and M.G. Sideris1 1Department of Geomatics Engineering, University of Calgary

2 Department of Earth Observation and Space Systems, Delft University of Technology

e-mail: [email protected]

Abstract. Satellite altimetry offers opportunities for studying crustal motion of regional and local origins. This is

achieved by the precise determination of the rate of change of the sea level from altimetry data combined with tide

gauge records of relative sea level change. In this study, we investigate an application of a wavelet-based

multiresolution analysis of TOPEX/Poseidon altimetry data (from 1992 to 2002) and tide gauge records to evaluate

trends of sea level change in the Baltic Sea. We compare the sea level trends with those estimated via a least-squares

(LS) regression approach. The conducted numerical experiments demonstrate that, in contrast to the wavelet-based

approach, LS regression tends to underestimate the sea level trends when short and noisy time series are analyzed. In

the presence of an anomalous sea level signal, the wavelet-based approach can detect and model this signal and also

accounts for this effect on the sea level trend.

1. Introduction

Today, satellite altimetry over oceans is being considered as complementary to the space positioning

techniques over land (SLR, GPS, DORIS, and VLBI) which, in combination with tide gauges, can be

employed in vertical crustal motion studies (see, e.g., Cazenave et al., 1999; Nerem and Mitchum, 2002).

While altimetry determines sea level in a geocentric coordinate system, tide gauges are attached to the

deformable solid Earth and measure the sea level with respect to it. Differences of the rate of change of

the two signals at the tide gauge location determine the absolute uplift (or subsidence), h , of the Earth’s

crust as follows:

Ssh −= , (1)

where s is the rate of absolute change of the sea level determined by altimetry and S is the rate of change

of relative sea level at the tide gauge. Eq. (1) requires that the altimeter instrument drift be precisely

known and the periodic variability in the altimetric and tide gauge sea level signals be either removed or

cancelled out. Differences between altimetric sea surface heights and tide gauge sea levels have been

analyzed for the purpose of postglacial rebound studies by, e.g., Shum et al., (2002) and Kuo et al., (2004)

and for vertical datum monitoring by Jekeli and Dumrongchai (2003).

2

In semi-enclosed sea regions such as the Baltic Sea, long-term oscillations in the altimetry signal

cannot be averaged out when linear trends are sought in contrast to global studies of sea level rise (see

Nerem and Mitchum, 2001). In an analysis of the Stockholm tide gauge record, Ekman and Stigebrandt

(1990) pointed out that the sea level in the Baltic Sea exhibits low-frequency atmospherically forced

oscillations. Such oscillations, and the presence of non-linear trends in the sea level signal, may obscure

the estimation of the rate of change of the sea level using least-squares (LS) regression (see also Barbosa

et al., 2004). Moreover, the absence of accurate tidal corrections for the Baltic Sea can be a critical factor

for the estimation of trends from short time span altimetry data since short-periodic oscillations may not

average out. If differences between altimetric and tide gauge sea level signals are utilized, some of the

periodic components may cancel out. However, to apply this method, only the last decade of the historic

tide gauge records can be used. To overcome this shortcoming, Kuo et al. (2004) used more than 40-year-

long tide gauge records in a Gauss-Markov model constrained by monthly sea level differences from

averaged altimetry sea surface heights and tide gauge records.

The aim of this study is to investigate the capabilities of the wavelet-based multiresolution analysis to

identify and remove the periodic components from the sea level signal prior to estimating the sea level

trend. Since wavelets have good localizing properties both in the time and scale domains, periodic

components present in the signal can be detected. This is achieved by imposing restrictions, i.e., threshold

values, on the detailed coefficients at different levels of decomposition. The numerical experiment in this

study has been conducted in a way that only the smoothing part of Mallat’s algorithm (Mallat, 1998) is

applied by suppressing the effect of detail coefficients from all levels. Under this condition, wavelets have

been used to define the low-pass filter (scaling functions) of Mallat’s algorithm, and we expect that

identical results should be obtained by means of other low-pass filtering techniques. By filtering out the

short-periodic components and using the smoothest time series obtained at the last possible level of

decomposition, the wavelet multiresolution analysis can be tested and compared to the traditional LS

regression technique in which a trend line is fitted to the time series. Both shorter altimetric and longer

tide gauge time series in the Baltic Sea are analyzed herein.

In the following, the methodology and the data sets used are described. Next, the investigated method

is tested using a synthetic sea level signal. This is followed by an analysis of the altimetric and tide gauges

time series. A discussion on the capabilities of the proposed method based on the wavelet multiresolution

analysis to model linear sea level trends is presented in the last section.

2. Description of the Methodology and Data Used

Wavelet-based multiresolution analysis (MRA) is the tool employed to numerically decompose and

reconstruct a signal. MRA helps in constructing filter banks based on dyadic wavelets (see Mallat, 1998),

3

and it consists of an application of low- and high-pass filters together with down-sampling and up-

sampling. The MRA smoothing step (scale functions) can be defined using different types of wavelets.

The general relation

ϕ

=ψ

2ˆ

2ˆ

2

1)(ˆ

ffhf (2)

between Fourier transforms of scale functions and the corresponding dyadic wavelets is used. Once the

dyadic wavelet is chosen, the corresponding scale function can be defined based on:

ϕ

=ϕ

2ˆ

2ˆ

2

1)(ˆ

fflf , (3)

where )(ˆ),(ˆ ff ϕψ are the Fourier transforms of the wavelet and scale functions;

2ˆ,

2ˆ f

lf

h are the

Fourier transforms of the high-pass and low-pass filters at twice finer resolution; and f is the frequency.

For the numerical experiment Daubechies 8 wavelets have been used and, following Keller (2000), the

corresponding scaling coefficients can be determined in the subsequent steps:

Step 1: Represent the scaling function in the form Ζ∈

−ϕ=ϕk

k kxlx )2(2)( , where scaling coefficients

kl need to be determined.

Step 2: Define a trigonometric polynomial

,))12(2cos(2

1)(

1

=

π−+=N

k

kN fkafq =

=N

k

ka1 2

1 assuming that 8=N .

Step 3: Find the spectrum of the scaling function ( )fϕ based on ( ) ( )fqf N=ϕ2

ˆ .

Step 4: Represent the spectrum of scaling function in terms of scaling coefficients

=

π−=ϕN

k

fjkkelf

0

2

2

1)(ˆ .

Step 5: Determine the scaling coefficients kl based on the assumed trigonometric polynomial and the

relation in step 3. For the reconstruction step, wavelet coefficients are assumed to be equal to zero.

The scaling coefficients depend on the choice of the trigonometric polynomial, which has a specific

form for Daubechies 8 wavelets (Mallat, 1998). Once the scaling coefficients are determined, the

smoothing part of Mallat’s algorithm can be applied.

MRA can be represented by the diagram in Fig.1 as decomposition and reconstruction steps of a signal.

The detail coefficients 1+jd are results of the high-pass filter while the approximated coefficients 1+ja

4

represent the smoothing part of the MRA at the (j+1)th level of decomposition. For the numerical

experiment in this study, only the smoothing part of the MRA is used due to the fact that all detail

coefficients are suppressed.

To exploit the good localization properties of the wavelets in the time and scale domains, a further

investigation of the evaluated different periodic components and the proper threshold values for every

level of decomposition is necessary. A complete application of wavelets for analyzing the periodic signals

present in altimeter and tide gauge time series will be discussed in a future paper.

The sea level trend is parameterized in the form of a second order polynomial with respect to the time t

as follows:

2ctbtaS ++= , (4)

where S is the time series and cba and,, are the polynomial coefficients determined via a LS fit to the time

series. From eq. (4), a linear trend is derived via differentiation with respect to time. This parameterization

is applied both to the last approximation level in the wavelet-based analysis and the original signal in the

LS regression method.

Fig.1 Decomposition and reconstruction steps in the wavelet-based multiresolution analysis (MRA).

l

h

2↓

2↓

ja 1+ja

1+jd

Decomposition step

l 2↓

h 2↓

2↑

2↑

l

h

1+ja 2+ja

2+jd

⊕

1+jd

Reconstruction step

2↑

2↑

h

l ⊕

2+ja

2+jd

5

We use TOPEX/Poseidon (T/P) time series (from 1992 to 2002 with approximately a 10-days time

resolution) at the crossover points in the Baltic Sea available from the AVISO GDR-M products (AVISO,

1996). The high coherence of the sea level signal at frequencies lower than one month has been used to

reconstruct the series at missing epochs (Lambeck et al., 1998). Due to the ice coverage at latitudes higher

than 63 degrees some of the crossover time series are discontinuous and no reliable interpolation for the

missing epochs can be done. In the conducted numerical experiments, we have used crossover time series

which, because of the reduced noise, are more appropriate for testing the wavelet-based analysis than the

time series obtained from along-track sea surface heights. The crossover adjustment aims to minimize the

radial orbit errors, unmodeled tides and other changes in the sea surface due to dynamic ocean factors.

Therefore, it is likely that the crossover adjustment reduces long-periodic variations and trends, and we

can expect that the sea level trend determined at the crossovers will be underestimated.

For the purpose of this study, tide gauge records relative to the Revised Local Reference (RLR) datum

at each site have been obtained from the Permanent Service for Mean Sea Level (PSMSL). To estimate

the secular trends, usually time series of annual means are used (see, e.g., Ekman and Stigebrandt, 1990).

However, they do not provide a sufficient resolution for the wavelet analysis, especially if shorter time

series are analyzed. Therefore, monthly mean values are used herein. No attempt to fill gaps larger than 3

months has been made. This fact and the space coverage of the altimetry data (54°<ϕ<63° and

14°<λ<26°) limit the number of the analyzed tide gauge series to sixteen.

3. Internal Validation of the Wavelet Multiresolution Analysis

First, the procedure has been validated by analyzing a synthetic signal. This signal contains periodic

variations estimated from the Stockholm tide gauge time series chosen to represent the sea level variability

of the entire Baltic Sea. The Stockholm tide gauge record exhibits small high frequency signal combined

with large long-term sea level variations (Ekman, 1999). Monthly and annual mean sea levels are

commonly used in studies of the seasonal variation in the sea level as well as the pole tide for the region

(Ekman and Stigebrandt, 1990; Wróblewski, 2001).

The synthetic signal includes: (i) a small trend of 0.3 mm/yr; (ii) a component due to the nodal tide

using estimates at the Stockholm tide gauge; (iii) a 6.5-year tidal component due to the overlapping of the

pole and annual tides; (iv) annual, semi-annual, approximately bi-, and tri-monthly periodic components

observed in the spectrum of the sea level signal; and (v) stationary random noise with variance of 1 mm2.

Since approximately a half period of the nodal tide component (with a period of 18.6 years) is present in

the altimetry signal, the trend can be parameterized as in eq. (4) from which a linear trend of 0.43 mm/yr

should be estimated (0.13 mm/yr is due to the nodal tide).

6

The signal has been analyzed using the Daubechies-8 wavelet (see Fig.2) and the trend has been

computed from the second order polynomial fitted to the approximation at the 8th level of decomposition.

Next, the trend has been estimated by LS regression. Table 1 compares the results obtained from 1000

runs of both tests at the assumed noise level. From the comparison between the mean, maximum, and

minimum trend given in the table, it can be seen that LS regression underestimates the trend while, after

filtering out the higher frequency components, the MRA correctly estimates the required trend to be 0.43

mm/yr.

0 1 2 3 4 5 6 7 8 9

-10

0

10

mm

0 1 2 3 4 5 6 7 8 9-5

0

5

0 1 2 3 4 5 6 7 8 9-5

0

5

mm

0 1 2 3 4 5 6 7 8 9-5

0

5

0 1 2 3 4 5 6 7 8 9-5

0

5

mm

years

0 1 2 3 4 5 6 7 8 9-5

0

5

years

min

-5.2

max

6.2

ave

0.0

std

2.2

min

-1.6

max

1.8

ave

0.0

std

0.7

min

-1.6

max

1.4

ave

0.0

std

0.7

min

-0.3

max

0.2

ave

0.0

std

0.1

min

-0.3

max

0.2

ave

0.0

std

0.1

min

-1.7

max

2.3

ave

0.2

std

1.4

level 5 level 6

level 7 level 8

level 4 level 3

Fig.2 Successive approximations of the synthetic signal and the residuals between them. The trend is estimated at the last level eight of decomposition. Note the different scale of the Y axis for level 3.

Table1 Comparison of the trend (in mm/yr) estimated via the wavelet-based analysis and LS regression for the synthetic time series.

mean st dev min max

Wavelets 0.43 ± 0.03 0.31 0.51

LS Regression 0.37 ± 0.02 0.30 0.42

7

4. Analysis of the Altimetric and Tide Gauge Time Series

The altimetric time series have been analyzed using the same Daubechies-8 wavelet. Fig.3a shows the

decomposition and successive approximations of one crossover time series. Fig.3b represents the spectra

of the residuals between the successive approximations. Oscillations with a frequency of approximately

two months are completely eliminated at level 3, while the semi-annual signal is removed at level 4. At

level 5, most of the annual signal is detected and eliminated. Finally, at the last level (8), the oscillations

due to the overlapping of the nodal and pole tides are removed, and the approximation represents the small

amplitude nodal tide, possibly, plus a trend component. The trend estimated from the last-level

approximation is 0.4 mm/yr while the LS regression trend is 0.1 ± 0.5 mm/yr. Table 2 compares the

estimated trend for all crossover time series. Apparently, the sea level trend estimated after removing the

short periodic variations is larger than the LS estimated trend. This agrees with the result obtained for the

synthetic time series. The average trend in the altimetric time series is 0.6±0.1 mm/yr and is less than the

mean sea level trend determined from satellite altimetry data (see, e.g., Cabanes et al., 2001). The

crossover adjustment is the most likely reason for that since it can reduce the amplitudes of long-term

variations and trends.

0 1 2 3 4 5 6 7 8 9

-10

0

10

mm

0 1 2 3 4 5 6 7 8 9

-10

0

10

0 1 2 3 4 5 6 7 8 9-5

0

5

mm

0 1 2 3 4 5 6 7 8 9-5

0

5

0 1 2 3 4 5 6 7 8 9-5

0

5

mm

years

0 1 2 3 4 5 6 7 8 9-5

0

5

years

min

-1.7 max

2.0

ave 0.1

std

1.2

min

-0.8 max

0.9

ave 0.1

std

0.5

min -1.3

max 1.3

ave

0.0 std

0.6

min -3.4

max 3.9

ave

0.0 std

1.1

min -8.3

max

8.4

ave 0.1

std

3.3

min -10.0

max

9.9

ave 0.0

std

3.8

Wavelet trend : 0.4 mm/yr

LS trend : 0.1 mm/yr +-0.5

level 3 level 4

level 5 level 6

level 7

Fig.3a Example of the successive approximations of altimetry crossover time series and statistics of the residuals.

0 1 2 3 4 5 60

1

2

mm

0 1 2 3 4 5 60

1

2

mm

0 1 2 3 4 5 60

1

2

mm

0 1 2 3 4 5 60

1

2

mm

0 0.5 1 1.5 2 2.5 30

1

2

cycles/yr

mm

0 0.5 1 1.5 2 2.5 30

1

2

cycles/yr

mm

level 3 level 4

level 5 level 6

level 7 level 8

Fig.3b Spectra of the residuals between the successive approximations of the altimetry time series.

Table 2 Comparison of the linear trends (in mm/yr) estimated via wavelets and LS regression for the crossovers.

1 2 3 4 5 6 7

Wavelets 0.36±0.00 0.50±0.01 0.22±0.02 0.50±0.00 0.68±0.01 0.44±0.00 0.67±0.01

LS Regression 0.12±0.55 0.15±0.64 0.06±0.80 0.17±0.50 0.34±1.0 0.15±0.50 0.26±0.90

8

Table 3 Secular trend identified in the tide gauge time series. According to the sign convention, the uplift (positive values) and subsidence (negative values) of the Earth’s crust are shown.

Ekman 1892-1991 1892 -2001 1950-2001 (1996) Wavelets LS Fitting Wavelets LS Fitting Wavelets LS Fitting

Klagshamn 0.04 - - - - -0.5 ± 0.2 -0.5 ± 1.8 Kungsholmsfort 0.20 0.2 ± 0.1 0.2 ± 0.9 0.1 ± 0.1 0.1 ± 0.7 -0.9 ± 0.2 -0.7 ± 2.1

Ölands Norra Udde 1.29 1.2 ± 0.1 1.2 ± 0.9 1.1 ± 0.1 1.2 ± 0.8 0.7 ± 0.2 0.9 ± 2.4 Landsort 3.06 3.0 ± 0.1 3.0 ± 0.9 2.9 ± 0.1 3.0 ± 0.8 2.1 ± 0.2 2.2 ± 2.5

Stockholm 4.00 4.0 ± 0.1 4.0 ± 0.9 3.9 ± 0.1 4.0 ± 0.8 3.4 ± 0.2 3.5 ± 2.5 Ratan 8.16 7.8 ± 0.1 7.9 ± 1.0 7.8 ± 0.1 7.9 ± 0.9 7.3 ± 0.2 7.4 ± 2.9

Furuögrund 8.75 - - - - 7.3 ± 0.2 7.5 ± 2.9 Pietarsaari/Jakobsta 8.04 - - - - 6.4 ± 0.2 6.6 ± 2.9

Kaskinen/Kasko 7.11 - - - - 6.0 ± 0.2 6.1 ± 2.8 Mantyluoto 6.31 - - - - 5.3 ± 0.2 5.4 ± 2.7

Rauma/Raumo 5.22 - - - - 4.4 ± 0.2 4.5 ± 2.7 Turku/Abo - - - - 3.0 ± 0.2 3.1 ± 2.8

Helsinki 2.28 2.4 ± 0.1 2.4 ± 1.0 2.2 ± 0.1 2.3 ± 0.9 1.0 ± 0.2 1.2 ± 2.9 Gdansk - - - - - -3.4 ± 0.2 -3.0 ± 2.5

Ustka - - - - - -2.1 ± 0.2 -1.8 ± 2.4 Warnemünde 2 -1.06 -1.2 ±0.1 -1.3 ± 0.5 -1.3 ±0.1 -1.2 ± 0.4 -1.1 ± 0.2 -1.4 ± 1.4

The tide gauge records have been first analyzed for the time interval from 1892 to 1991. The last

decade of the 20th century, which coincides with the time span of the altimetric measurements, is

characterized by low-water years and usually is excluded from analyses of the sea level rates for the

region (Lambeck et al., 1998). In Table 3, our estimates and those published by Ekman (1996) for the

same time interval but using annual mean values are compared. According to the adopted sign convention,

the table represents the relative uplift/subsidence of the Earth’s surface with respect to the sea level. The

differences between Ekman’s trend estimates and the ones estimated in this study (using monthly means)

is within the formal error of the LS regression. Furthermore, both wavelets and LS regression give very

close estimates, which demonstrates that for long time series and when no anomalous sea level signal is

present wavelets and LS regression give identical results. Latter in this section, we demonstrate that

wavelets are capable of modeling the anomalous signal present in the sea level time series. In the case of

shorter time series, this leads to larger estimated trends.

Figures 4a and 4b show the decomposition and the spectra of the residuals between the approximations

of the sea level signal at the Stockholm tide gauge. The annual and semi-annual signals are removed at

level 3, while long-term oscillations are still present in the approximation at level 8. Similarly to the

altimetric time series, a second order trend curve is fitted to the last-level approximation. From Table 3, it

is evident that both the wavelet-based analysis and LS regression estimate a trend of 4.0 mm/yr.

The much smaller error for wavelets is due to the smaller residuals after filtering out high frequency

oscillations from the signal.

The addition of data from the last decade of the 20th century to the longest continuous records reduces

the secular trend by 0.1 mm/yr at most of the sites (see Table 3), but this difference is still within the

9

estimation error. If the time series from 1950 to 2001 (from 1951 to 1999 for the tide gauge sites Gdansk

and Ustka) are analyzed, the estimated trend is lowered by 0.8 mm/yr on average. For the tide gauges on

the south coast of the Baltic Sea, where smaller rates are observed, the differences between the wavelet-

based analysis and the LS regression become larger than the formal errors. For the tide gauge Gdansk, LS

regression predicts a smaller rate of sea level rise compared to the wavelet analysis (see Fig. 5). Because

of the good localizing properties of wavelets in time domain, the low-water signal in the last decade of the

record (see the box in Fig.5) is detected and modeled. The polynomial curve fitted to the last-level

approximation accounts for this local effect and the estimated trend is larger. This is observed also for

Ustka and Kungsholmsfort, but not for Klagshamn and Warnemünde. Because the first two tide gauge

sites have larger estimation errors (see Table 3), it is not clear whether the larger trend estimated by the

wavelet-based analysis is realistic. Additional numerical experiments and analysis are necessary in order

to establish the advantages of using wavelets in the presence of an anomalous sea level signal especially in

the case of short and noisy time series.

0 10 20 30 40 50 60 70 80 90

-800

-600-400

-200

0

200

mm

0 10 20 30 40 50 60 70 80 90-600

-400

-200

0

200

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-200

0

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mm

0 10 20 30 40 50 60 70 80 90-600

-400

-200

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200

0 10 20 30 40 50 60 70 80 90-600

-400

-200

0

200

mm

years

0 10 20 30 40 50 60 70 80 90-600

-400

-200

0

200

years

min

-24.4

max

26.4

ave

-0.6

std

12.6

min

-11.6

max 11.5

ave

-0.3

std

5.4

min

-116.6

max

94.9

ave

-0.4

std

32.6

min

-143.9

max

129.7

ave

0.4

std

36.1

min

-141.4

max

143.5

ave

0.2

std

43.5

min

-494.1

max

390.4

ave

0.0

std

140.7

level 4

level 5 level 6

level 7 level 8

level 3

Fig.4a Successive approximations of the Stockholm tide gauge time series and statistics of the residuals. Note the different scales of the Y axis.

0 1 2 3 4 5 60

10

20

30

mm

0 1 2 3 4 5 60

10

20

30

mm

0 1 2 3 4 5 60

10

20

30

mm

0 1 2 3 4 5 60

10

20

30m

m

0 0.5 1 1.5 20

10

20

30

cycles/yr

mm

0 0.5 1 1.5 20

10

20

30

mm

cycles/yr

level 3 level 4

level 5 level 6

level 7 level 8

93 mm

40 mm

Fig.4b Spectra of the residuals between the successive approximations of the time series.

10

0 5 10 15 20 25 30 35 40 45 50-400

-300

-200

-100

0

100

200

300

400

500

600

years

mm

approximation

quadratic trend from the appr

LS regression

TG record

Fig. 5 Sea level trend determined by the wavelet-based analysis and LS regression from the tide gauge time series at Gdansk (1951-1999).

6. Discussion and Conclusions

To compute the absolute rates of the uplift/subsidence, the sea level trend estimated at the altimetry

crossovers must be interpolated and added to the sea level rates relative to the Earth’s surface according to

the sign convention in Table 3. No such attempts have been made in this study. No conclusions about the

absolute crustal uplift/subsidence can be drawn based on an analysis performed at a very limited number

of crossover points across the Baltic Sea. On the other hand, time series of sea surface heights averaged

around tide gauge sites can be created and used in the wavelet-based analysis. Nevertheless, this numerical

experiment has demonstrated that LS regression tends to underestimate the sea level trend when short

and/or noisy time series are analyzed. In the presence of an anomalous sea level signal, the wavelet-based

approach can detect and model this signal and accounts for this effect on the trend.

In order to draw more definite conclusions about the application of the wavelet analysis for estimating

sea level trends, a comparison with other smoothing methods is necessary. One such method is the robust

non-parametric smoothing based on locally weighted regression; see Barbosa et al. (2004). As pointed out

by the authors, this non-parametric method offers more flexibility in describing non-linear trends than LS

regression, and can be very effective in analyzing short sea level time series.

As discussed herein, the results obtained must be considered as a preliminary step in our attempts to

apply the wavelet-based multiresolution analysis in order to determine sea level trends from altimetry data

for the region of the Baltic Sea. Based on these results, there exist indications that filtering out high

11

frequency oscillations (partly oceanographic signal) from the short-term sea level records can improve the

estimated trend. If proven to provide reliable estimates for sea level trends, the suggested procedure can be

incorporated in the determination of the absolute rates of vertical crustal motion associated with

postglacial rebound in the region.

Acknowledgments

We thank AVISO and PSMSL for providing the data. The anonymous reviewers are gratefully

acknowledged for their careful comments and suggestions that improved the quality of this paper. This

work is supported by GEOIDE NCE and NSERC grants to the third author.

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